Stability and chaos in real polynomial maps

Thesis presented by
Fermín Franco-Medrano, B.Sc.
As a requirement to obtain the degree of
Master of Science with specialty in Applied Mathematics
conferred by
Centro de Investigación en Matemáticas, A.C.
Thesis Advisor:
Francisco Javier Solís Lozano, Ph.D.
[Uncaptioned image] Guanajuato, GT, United Mexican States, June 2013.

Stability and chaos in real polynomial maps

Thesis presented by
Fermín Franco-Medrano, B.Sc.
As a requirement to obtain the degree of
Master of Science with specialty in Applied Mathematics
conferred by
Centro de Investigación en Matemáticas, A.C.
Approved by the Thesis Advisor:
 
Francisco Javier Solís Lozano, Ph.D.
Guanajuato, GT, United Mexican States, June 2013.

Abstract

We extend and improve the existing characterization of the dynamics of general quadratic real polynomial maps with coefficients that depend on a single parameter λ𝜆\lambda, and generalize this characterization to cubic real polynomial maps, in a consistent theory that is further generalized to real n𝑛n-th degree real polynomial maps. In essence, we give conditions for the stability of the fixed points of any real polynomial map with real fixed points. In order to do this, we have introduced the concept of Canonical Polynomial Maps which are topologically conjugate to any polynomial map of the same degree with real fixed points. The stability of the fixed points of canonical polynomial maps has been found to depend solely on a special function termed Product Distance Function for a given fixed point. The values of this product distance determine the stability of the fixed point in question, when it bifurcates, and even when chaos arises, as it passes through what we have termed stability bands. The exact boundary values of these stability bands are yet to be calculated for regions of type greater than one for polynomials of degree higher than three.

Acknowledgements

I have both the duty and pleasure of thanking the support of many people and institutions that helped me in the process of conducting my graduate studies. In the first place, I greatly thank the support of the National Council of Science and Technology of the United Mexican States (CONACYT) for their financial support as a national scholar during my studies. I thank my teachers and the personnel of the Center for Mathematical Research (CIMAT) in Guanajuato, for their patient and committed work from which I nurtured and which allowed me to grow in this Mexican institution. In particular, I thank the orientation and guidance of Professor Francisco Javier Solís Lozano, my thesis advisor; as well as Professors Lázaro Raúl Felipe Parada, Mónica Moreno Rocha and Daniel Olmos Liceaga, who contributed greatly to the improvement of this work with their comments and advice; and Dr. Marcos Aurelio Capistrán Ocampo for his support and orientation. I also thank Mr. José Guadalupe Castro López and my friends and classmates for their support and invaluable company. To my family and particularly to my mother, I must extend the greatest gratitude of all.

Chapter 1 Introduction

In this chapter, we will briefly address the basic concepts of the theory of discrete dynamical systems, so as the concept of chaos that can occur in these systems, as much as necessary to define the relevant problem and the goals of this thesis. Important references in this subject are [Devaney(1989), Holmgren(1994), Elaydi(2000)], among others, and all of the proofs omitted here can be found therein. This chapter is included with the aim of making the work self-contained, it includes only what is now the standard basic theory of discrete dynamical systems, and may be skipped if the reader is familiar with it.

This chapter has the following objectives: (i) familiarize with the fundamentals of the theory of discrete dynamical systems, particularly chaotic ones; (ii) show the importance of the parametric dependency in families of functions, particularly bifurcation theory; and (iii) present the Feigenbaum sequence and its universality property for unimodal maps as a basis to understand how chaos may arise through period doubling bifurcations.

1.1 Elementary concepts

The main objective of the theory of discrete dynamical systems is to understand the final or asymptotic behavior of an iterative process. If the process is continuous, a differential equation with time being the independent variable, then the theory intends to predict the ultimate behavior of the solutions in the distant future (t𝑡t\rightarrow\infty) or the distant past (t𝑡t\rightarrow-\infty). If the process is discrete, such as the iteration of a real (or complex) function f𝑓f, then the theory hopes to be able to understand the eventual behavior of the points x,f(x),f2(x),,fn(x)𝑥𝑓𝑥superscript𝑓2𝑥superscript𝑓𝑛𝑥x,f(x),f^{2}(x),...,f^{n}(x) when n𝑛n\in\mathbb{N} is large, where fn(x)superscript𝑓𝑛𝑥f^{n}(x) denotes the n𝑛n-th iteration (i.e., composition of the function with itself). That is, the theory of dynamical systems asks the question: where do points go and what do they do when they get there? In this work, we aim to answer a part of this question for one of the simplest kinds of dynamical systems: functions of a single real variable; in particular, for real polynomial functions, having as a precedent the work of [Solís and Jódar(2004)] for quadratic maps.

In the following, let A𝐴A\subseteq\mathbb{R}, xA𝑥𝐴x\in A and f:A:𝑓𝐴f:A\rightarrow\mathbb{R}.

Definition 1.1 (Orbits).

The forward orbit of x𝑥x is the set of points x,f(x),f2(x),𝑥𝑓𝑥superscript𝑓2𝑥x,\,f(x),\,f^{2}(x),\,... and is denoted by O+(x)superscript𝑂𝑥O^{+}(x). If f𝑓f is a homeomorphism, we can define the complete orbit of x𝑥x, O(x)𝑂𝑥O(x), as the set of points fn(x)superscript𝑓𝑛𝑥f^{n}(x) for n𝑛n\in\mathbb{Z}, and the backward orbit of x𝑥x, O(x)superscript𝑂𝑥O^{-}(x), as the set of points x,f1(x),f2(x),𝑥superscript𝑓1𝑥superscript𝑓2𝑥x,\,f^{-1}(x),\,f^{-2}(x),\,....

Therefore, we can restate the basic goal of discrete dynamical systems as to know and understand all orbits of a map. Orbits can be quite complicated sets, even for simple non-linear maps. However, there are some specially simple orbits which play a central role in the dynamics of the whole system.

Definition 1.2 (Periodic Points).

The point x𝑥x is a fixed point of f𝑓f if f(x)=x𝑓𝑥𝑥f(x)=x. The point x𝑥x is a periodic point of period n𝑛n if fn(x)=xsuperscript𝑓𝑛𝑥𝑥f^{n}(x)=x. The smallest positive n𝑛n for which fn(x)=xsuperscript𝑓𝑛𝑥𝑥f^{n}(x)=x is called the prime period of x𝑥x. We denote the set of periodic points of period n𝑛n (not necessarily prime) by Pern(f)subscriptPer𝑛𝑓\mathrm{Per}_{n}(f), and the set of fixed points by Fix(f)Fix𝑓\mathrm{Fix}(f). The set of all iterates of a periodic point forms a periodic orbit.

An important question is the “range of influence” of a fixed point. That is to ask, what set of points, if any, will approach or tend to a fixed (or periodic) point under the iteration of f𝑓f.

Definition 1.3 (Stable Set or Basin of Attraction).

Let p𝑝p be periodic of period n𝑛n. A point x𝑥x is forward asymptotic to p𝑝p if limifin(x)=psubscript𝑖superscript𝑓𝑖𝑛𝑥𝑝\lim_{i\rightarrow\infty}f^{in}(x)=p. The stable set (or basin of attraction) of p𝑝p, denoted by Ws(p)superscript𝑊𝑠𝑝W^{s}(p), consists of all points forward asymptotic to p𝑝p.

If p𝑝p is not periodic, we can still define its forward asymptotic points requiring that |fi(x)fi(p)|0superscript𝑓𝑖𝑥superscript𝑓𝑖𝑝0|f^{i}(x)-f^{i}(p)|\rightarrow 0 when i𝑖i\rightarrow\infty. Moreover, if f𝑓f is invertible, we can consider backward asymptotic points by taking i𝑖i\rightarrow-\infty in the last definition. The set of backward asymptotic points to p𝑝p is called the unstable set of p𝑝p and is denoted by Wu(p)superscript𝑊𝑢𝑝W^{u}(p).

Another useful concept comes from elementary calculus.

Definition 1.4 (Critical Points).

A point x𝑥x is a critical point of f𝑓f if f(x)=0superscript𝑓𝑥0f^{\prime}(x)=0. The critical point is non degenerate if f′′(x)0superscript𝑓′′𝑥0f^{\prime\prime}(x)\neq 0. The critical point is degenerate if f′′(x)=0superscript𝑓′′𝑥0f^{\prime\prime}(x)=0.

Critical points will prove later to be important in relation to the stable set of a fixed point, and to determine how many attracting orbits a map may have.

Degenerate critical points can be either maxima, minima or saddle points. Non degenerate critical points can only be maxima or minima. A diffeomorphism cannot have critical points, but its existence in non-invertible maps is one of the main reasons why this type of maps are more complicated.

The goal of the theory of discrete dynamical systems is then to understand the nature of all orbits and to identify which are periodic, asymptotic, etc. In general, this is an impossible task. For example, if f(x)𝑓𝑥f(x) is a quadratic polynomial, then explicitly finding the periodic points of period n𝑛n is equivalent to solving the equation fn(x)=xsuperscript𝑓𝑛𝑥𝑥f^{n}(x)=x, which is a polynomial equation of degree 2nsuperscript2𝑛2^{n}. Computer calculations do not help in this case either, since rounding errors tend to accumulate and make many periodic points “invisible” to the computer. Therefore, qualitative and geometrical techniques must be employed to understand the nature of a system.

1.2 Hyperbolicity and criteria for stability

A key issue once we find periodic points, is to determine the behavior of orbits near these points; such investigation is what is called stability theory. Once this is done, we can state whether a periodic point is attracting or repelling (or neither). To determine this, we can part from the following

Definition 1.5 (Stability).

Let pA𝑝𝐴p\in A be a fixed point of f𝑓f. Then,

  1. 1.

    p𝑝p is said to be stable if for any ε>0𝜀0\varepsilon>0 there exists a δ>0𝛿0\delta>0 such that if |x0p|<δsubscript𝑥0𝑝𝛿|x_{0}-p|<\delta then |fn(x0)p|<εsuperscript𝑓𝑛subscript𝑥0𝑝𝜀|f^{n}(x_{0})-p|<\varepsilon for all integers n+𝑛superscriptn\in\mathbb{Z}^{+}. Otherwise, the fixed point p𝑝p is called unstable.

  2. 2.

    p𝑝p is called attracting if there is a η>0𝜂0\eta>0 such that if |x0p|<ηsubscript𝑥0𝑝𝜂|x_{0}-p|<\eta then limnfn(x0)=psubscript𝑛superscript𝑓𝑛subscript𝑥0𝑝\lim_{n\rightarrow\infty}f^{n}(x_{0})=p.

  3. 3.

    p𝑝p is asymptotically stable if it is both stable and attracting.

Therefore, intuitively, a fixed point is stable if nearby orbits remain near the fixed point, and you can always find an interval around the fixed point such that orbits starting within the interval will remain arbitrarily close to the fixed point. The opposite is that you cannot do that; i.e. there is an interval around the fixed point such that no matter how close to the fixed point you start the orbit from, the orbit will go outside the interval. The case of an asymptotically fixed point is particularly important, since in this case all the orbits near a fixed point will approach it in the limit. The next step is to be able to determine when a fixed point is stable or unstable. To do this, we will need the

Definition 1.6 (Hyperbolic Point).

Let p𝑝p be a periodic point of period n𝑛n. The point p𝑝p is hyperbolic if |(fn)(p)|1superscriptsuperscript𝑓𝑛𝑝1|(f^{n})^{\prime}(p)|\neq 1. Otherwise, it is called a nonhyperbolic fixed point. The number (fn)(p)superscriptsuperscript𝑓𝑛𝑝(f^{n})^{\prime}(p) is called the multiplier of the periodic point.

Example 1.1.

Consider the diffeomorphism f(x)=12(x3+x)𝑓𝑥12superscript𝑥3𝑥f(x)=\frac{1}{2}(x^{3}+x). There are three fixed points: x=0, 1,𝑥01x=0,\,1, and -1. Note that f(0)=1/2superscript𝑓012f^{\prime}(0)=1/2 and f(±1)=2superscript𝑓plus-or-minus12f^{\prime}(\pm 1)=2. Therefore, each point is hyperbolic.

As it turns out, hyperbolic points are easy to understand in terms of whether they are attracting or repelling, as they are always one of the two.

Definition 1.7 (Attractor and repellor).

Let p𝑝p be a hyperbolic point of period n𝑛n. If |(fn)(p)|<1superscriptsuperscript𝑓𝑛𝑝1|(f^{n})^{\prime}(p)|<1, the point p𝑝p is asymptotically stable and is called an attracting periodic point or more shortly, an attractor; if |(fn)(p)|>1superscriptsuperscript𝑓𝑛𝑝1|(f^{n})^{\prime}(p)|>1, p𝑝p is unstable and it is called a repelling periodic point or, simply, a repellor.

Occasionally, the terms sink and source may also be used to refer to attractors and repellors, respectively, stemming from the continuous dynamical systems analog. Now, if a periodic point is attracting, it must have an interval around it in which it is so; this is stated in the following

Theorem 1.1.

Let p𝑝p be a hyperbolic fixed point of f𝑓f. Then if p𝑝p is an attractor, there exists an open set UWs(p)𝑈superscript𝑊𝑠𝑝U\subseteq W^{s}(p), with pU𝑝𝑈p\in U; on the other hand, if p𝑝p is a repellor, there is an open set VWu(p)𝑉superscript𝑊𝑢𝑝V\subset W^{u}(p), with pV𝑝𝑉p\in V.

This theorem explains the choice of the terms “attracting” an “repelling”, since attractors and repellors have around them a basin of attraction and an unstable set, respectively.

It is straightforward to see that Ws(p)superscript𝑊𝑠𝑝W^{s}(p) is an invariant set under the action of the map. A similar result is true for periodic points of period n𝑛n. In this case, we have an open interval around p𝑝p that is mapped into itself by fnsuperscript𝑓𝑛f^{n}.

1.3 The Schwarzian derivative

Here, we will discuss a mathematical tool that will turn out useful in determining some properties of maps in terms of its periodic points structure.

Definition 1.8 (Schwarzian Derivative).

The Schwarzian derivative of a function f𝑓f at the point x𝑥x is given by

Sf(x)=f′′′(x)f(x)32(f′′(x)f(x))2.𝑆𝑓𝑥superscript𝑓′′′𝑥superscript𝑓𝑥32superscriptsuperscript𝑓′′𝑥superscript𝑓𝑥2Sf(x)=\frac{f^{\prime\prime\prime}(x)}{f^{\prime}(x)}-\frac{3}{2}\left(\frac{f^{\prime\prime}(x)}{f^{\prime}(x)}\right)^{2}.

In particular, it was first used by [Singer(1978)] in [Singer(1978)] to address the question of how many attracting periodic points a differentiable map can have [Singer(1978)] (see Singer’s Theorem below). It can also be used to determine the nature of non-hyperbolic periodic points [Elaydi(2000)]. Mappings with negative Schwarzian derivative present particular dynamical properties that will interest us in this work.

Theorem 1.2.

Let P(x)𝑃𝑥P(x) be a real polynomial. If all roots of P(x)superscript𝑃𝑥P^{\prime}(x) are real and distinct, then SP(x)<0𝑆𝑃𝑥0SP(x)<0 for all x𝑥x.

The following theorem is useful when analyzing topologically conjugate maps (a concept to be defined further below in 1.9), which in turn is useful to determine the stability properties of one map in terms of the known such properties of another map (the conjugate map).

Theorem 1.3.

Suppose Sf<0𝑆𝑓0Sf<0 and Sg<0𝑆𝑔0Sg<0. Then S(fg)<0𝑆𝑓𝑔0S(f\circ g)<0.

Also, this tells us that iteration of a map does not change the sign of its Schwarzian derivative, i.e. the property is preserved. This immediately takes us to

Corollary.

Let Sf<0𝑆𝑓0Sf<0. Then Sfn<0𝑆superscript𝑓𝑛0Sf^{n}<0 for all n>1𝑛1n>1.

The assumption of Sf<0𝑆𝑓0Sf<0 has several surprising implications for the dynamics of a one-dimensional map, as we will see below. Another useful result dealing with the geometry of a map is

Lemma 1.4.

If Sf<0𝑆𝑓0Sf<0, then f(x)superscript𝑓𝑥f^{\prime}(x) cannot have a positive local minimum or a negative local maximum.

The proof of this last lemma is straightforward from the definition of the Schwarzian derivative and elementary calculus. Note that this lemma tells us that, if Sf<0𝑆𝑓0Sf<0, between any two consecutive critical points of fsuperscript𝑓f^{\prime}, its graph must cross the x𝑥x-axis; this in turn means that there must be a critical point for f𝑓f between these two points, i.e. there must be a maximum or minimum of f𝑓f between any pair of its inflection points.

The Schwarzian derivative can also be used to determine whether a fixed point attracts a critical point of a map.

Lemma 1.5.

Let a1,a2subscript𝑎1subscript𝑎2a_{1},a_{2} and a3subscript𝑎3a_{3} be fixed points of a continuously differentiable map g𝑔g with a1<a2<a3subscript𝑎1subscript𝑎2subscript𝑎3a_{1}<a_{2}<a_{3} and such that Sg<0𝑆𝑔0Sg<0 on the open interval (a1,a3)subscript𝑎1subscript𝑎3(a_{1},\,a_{3}). If g(a2)1superscript𝑔subscript𝑎21g^{\prime}(a_{2})\leq 1, then g𝑔g has a critical point in (a1,a3)subscript𝑎1subscript𝑎3(a_{1},\,a_{3}).

The last lemma is used in the proof of the following important result due to [Singer(1978)] ([Singer(1978)]).

Theorem 1.6 (Singer’s Theorem).

Suppose Sf<0𝑆𝑓0Sf<0 (Sf(x)=𝑆𝑓𝑥Sf(x)=-\infty is allowed). Suppose f𝑓f has n𝑛n critical points. Then f𝑓f has at most n+2𝑛2n+2 attracting periodic orbits.

This last theorem allows to find directly from f𝑓f an upper bound for the number of attracting periodic points. This differs from the information provided in the analysis by Sarkovskii’s theorem (see section 1.5 below) in that the latter tells us how many periodic points there are, regardless of whether they are attracting or not and, also, we have first to find some periodic point and its period to be able to apply Sarkovskii’s theorem.

1.4 Nonhyperbolic fixed points

The stability criteria for nonhyperbolic fixed points involve the Schwarzian derivative defined in 1.3. We will analyze the cases of the multiplier being equal to 1 and -1 separately. The unstated proofs of the following theorems can be found in the book by [Elaydi(2000)].

Theorem 1.7.

Let p𝑝p be a fixed point of a map f𝑓f such that f(p)=1superscript𝑓𝑝1f^{\prime}(p)=1. Then, if f′′′(p)0superscript𝑓′′′𝑝0f^{\prime\prime\prime}(p)\neq 0 and is continuous, the following statements hold

  1. 1.

    If f′′(p)0superscript𝑓′′𝑝0f^{\prime\prime}(p)\neq 0, then p𝑝p is unstable.

  2. 2.

    If f′′(p)=0superscript𝑓′′𝑝0f^{\prime\prime}(p)=0 and f′′′(p)>0superscript𝑓′′′𝑝0f^{\prime\prime\prime}(p)>0, then p𝑝p is unstable.

  3. 3.

    If f′′(p)=0superscript𝑓′′𝑝0f^{\prime\prime}(p)=0 and f′′′(p)<0superscript𝑓′′′𝑝0f^{\prime\prime\prime}(p)<0, then p𝑝p is asymptotically stable.

Similarly, we have

Theorem 1.8.

Let p𝑝p be fixed point of a map f𝑓f such that f(p)=1superscript𝑓𝑝1f^{\prime}(p)=-1. If f′′′(p)superscript𝑓′′′𝑝f^{\prime\prime\prime}(p) is continuous, then the following statements hold:

  1. 1.

    If Sf(p)<0𝑆𝑓𝑝0Sf(p)<0, then p𝑝p is an asymptotically stable.

  2. 2.

    If Sf(p)>0𝑆𝑓𝑝0Sf(p)>0, then p𝑝p is unstable.

It is worth noting that a repellor is an unstable periodic point but not all such points are repellors since, a nonhyperbolic periodic point may also be unstable; e.g. a “semistable” nonhyperbolic fixed point is “attracting from the right (or left)” but “repelling from the left (or right, respectively)”.

When dealing with nonhyperbolic points, particularly when they are unstable, it is useful to account for the concept of “semistability”, separating stability “from the right” and “from the left”.

Definition 1.9 (Semistability).

A fixed point p𝑝p of a map f𝑓f is said to be semistable from the right (respectively, from the left) if for any ε>0𝜀0\varepsilon>0 there exists δ>0𝛿0\delta>0 such that if 0<x0p<δ0subscript𝑥0𝑝𝛿0<x_{0}-p<\delta (respectively, 0<px0<δ0𝑝subscript𝑥0𝛿0<p-x_{0}<\delta) then |fn(x0)p|<εsuperscript𝑓𝑛subscript𝑥0𝑝𝜀|f^{n}(x_{0})-p|<\varepsilon for all n+𝑛superscriptn\in\mathbb{Z}^{+}. Moreover, if limnfn(x0)=p𝑙𝑖subscript𝑚𝑛superscript𝑓𝑛subscript𝑥0𝑝lim_{n\rightarrow\infty}f^{n}(x_{0})=p whenever 0<x0p<η0subscript𝑥0𝑝𝜂0<x_{0}-p<\eta (respectively, 0<px0<η0𝑝subscript𝑥0𝜂0<p-x_{0}<\eta) for some η>0𝜂0\eta>0, then p𝑝p is said to be semiasymptotically stable from the right (respectively, from the left).

It is then easy to prove that

Theorem 1.9 (Asymptotical semistability).

Let f𝑓f be a map, p𝑝p one of its fixed points and with f(p)=1superscript𝑓𝑝1f^{\prime}(p)=1 and f′′(p)0superscript𝑓′′𝑝0f^{\prime\prime}(p)\neq 0. Then p𝑝p is

  1. 1.

    Semiasymptotically stable from the right if f′′(p)<0superscript𝑓′′𝑝0f^{\prime\prime}(p)<0;

  2. 2.

    Semiasymptotically stable from the left if f′′(p)>0superscript𝑓′′𝑝0f^{\prime\prime}(p)>0.

1.5 Sarkovskii’s theorem

This theorem is of great beauty, as it constructs a new –apparently artificial– ordering of the natural numbers in order to determine what types of orbits may a map have. Consider then the following ordering of the natural numbers:

357232527223225357232527superscript223superscript2253\triangleright 5\triangleright 7\triangleright\cdots\triangleright 2\cdot 3\triangleright 2\cdot 5\triangleright 2\cdot 7\triangleright\cdots\triangleright 2^{2}\cdot 3\triangleright 2^{2}\cdot 5\triangleright\cdots
233235232221.superscript233superscript235superscript23superscript2221\triangleright 2^{3}\cdot 3\triangleright 2^{3}\cdot 5\triangleright\cdots\cdots\triangleright 2^{3}\triangleright 2^{2}\triangleright 2\triangleright 1.

That is: first all odd numbers, then two times each odd number, 22superscript222^{2} times each odd number, 23superscript232^{3} times each odd number, etc., and finally, the powers of two in descending order. This is Sarkovskii’s ordering of the natural numbers and it easy to see it is exhaustive .

Having defined the above, we can state

Theorem 1.10 (Sarkovskii).

Let f::𝑓f:\mathbb{R}\rightarrow\mathbb{R} continuous. Suppose that f𝑓f has a periodic point of prime period k𝑘k. If kl𝑘𝑙k\triangleright l in Sarkovskii’s ordering, then f𝑓f also has a periodic point of period l𝑙l.

Thus, Sarkovskii’s theorem gives a nearly complete answer to the question of how many periodic points and of which periods can a map have, although it does not tell us their nature in terms of their stability, nor does it tell us how to find such points.

Some important consequences of Sarkovskii’s Theorem can be summarized as follows:

  • If f𝑓f has periodic points with periods different from a power of 2, then f𝑓f has infinite periodic points of infinitely different periods.

  • If f𝑓f only has a finite number of periodic points, then necessarily all points have periods of powers of 2.

  • Period 3 is the “greatest” period in Sarkovskii’s ordering, therefore, it implies the existence of all other periods.

Sarkovskii’s theorem is without doubt a central result for the theory of discrete dynamical systems.

1.6 Bifurcation theory

We will now consider families of maps instead of single maps, as this is often useful in applications and is also the main object of study of this work. For example, in practice, the coefficients of a polynomial map that models some natural, physical, economic or social discrete process can only be determined up to a certain precision, which gives rise to a family of maps as a set bounded by the uncertainties of the coefficients.

The aim of bifurcation theory is then to study the changes that occur in a map when its parameters are varied. This changes commonly refer qualitatively to its periodic points structure and stability properties, but they can also involve other changes. Consider families of functions in one real variable which depend upon a single real parameter (uniparametric). Consider the bivariate function G(x,λ)=fλ(x)𝐺𝑥𝜆subscript𝑓𝜆𝑥G(x,\lambda)=f_{\lambda}(x), where for fixed λ𝜆\lambda\in\mathbb{R}, fλ(x)Csubscript𝑓𝜆𝑥superscript𝐶f_{\lambda}(x)\in C^{\infty}. We will assume also that G𝐺G depends smoothly on λ𝜆\lambda. Some examples of such families of functions are:

  • Fμ(x)=μx(1x)subscript𝐹𝜇𝑥𝜇𝑥1𝑥F_{\mu}(x)=\mu x(1-x).

  • Eλ(x)=λexsubscript𝐸𝜆𝑥𝜆superscript𝑒𝑥E_{\lambda}(x)=\lambda e^{x}.

  • Sλ(x)=λsin(x)subscript𝑆𝜆𝑥𝜆𝑥S_{\lambda}(x)=\lambda\sin(x).

  • Qc(x)=x2+csubscript𝑄𝑐𝑥superscript𝑥2𝑐Q_{c}(x)=x^{2}+c.

With this in mind, we will next give a rather informal definition of bifurcation.

Definition 1.10 (Bifurcation).

Let fλ(x)subscript𝑓𝜆𝑥f_{\lambda}(x) be a uniparametric family of functions. Then, we say there is a bifurcation at λ0subscript𝜆0\lambda_{0} if there exists ϵ>0italic-ϵ0\epsilon>0 such that if a𝑎a and b𝑏b satisfy λ0ϵ<a<λ0subscript𝜆0italic-ϵ𝑎subscript𝜆0\lambda_{0}-\epsilon<a<\lambda_{0} and λ0<b<λ0+ϵsubscript𝜆0𝑏subscript𝜆0italic-ϵ\lambda_{0}<b<\lambda_{0}+\epsilon, then the “dynamics” of fa(x)subscript𝑓𝑎𝑥f_{a}(x) is different from the “dynamics” of fb(x)subscript𝑓𝑏𝑥f_{b}(x). That is, the “dynamics” of the function fλ(x)subscript𝑓𝜆𝑥f_{\lambda}(x) changes when λ𝜆\lambda passes through the value λ0subscript𝜆0\lambda_{0}.

The “changes in the dynamics” of the above definition refer to qualitative changes in the structure of periodic points of fλ(x)subscript𝑓𝜆𝑥f_{\lambda}(x). To make the above notion precise, we must consider separately the different types of bifurcations that may occur. For real maps, the saddle-node and period doubling bifurcations are the most common. We will next discuss the concept of saddle-node bifurcation through an example and the period doubling bifurcation is discussed in more detail below.

Example 1.2 (Saddle-node or Tangent Bifurcation).

Let’s consider the family of functions Eλ(x)=λexsubscript𝐸𝜆𝑥𝜆superscript𝑒𝑥E_{\lambda}(x)=\lambda e^{x}, with λ>0𝜆0\lambda>0. A bifurcation occurs when λ=1/e𝜆1𝑒\lambda=1/e. We have three cases:

  1. 1.

    If λ>1/e𝜆1𝑒\lambda>1/e the graph of Eλ(x)subscript𝐸𝜆𝑥E_{\lambda}(x) and y=x𝑦𝑥y=x do not intersect each other, so that Eλsubscript𝐸𝜆E_{\lambda} has no fixed points. Eλn(x)xsuperscriptsubscript𝐸𝜆𝑛𝑥for-all𝑥E_{\lambda}^{n}(x)\rightarrow\infty\,\forall x.

  2. 2.

    When λ=1/e𝜆1𝑒\lambda=1/e, the graph of Eλsubscript𝐸𝜆E_{\lambda} tangentially touches the identity diagonal at (1, 1)11(1,\,1). Eλ(1)=1subscript𝐸𝜆11E_{\lambda}(1)=1 is the only fixed point. If x<1𝑥1x<1, Eλn(x)1superscriptsubscript𝐸𝜆𝑛𝑥1E_{\lambda}^{n}(x)\rightarrow 1 and if x>1𝑥1x>1, Eλn(x)superscriptsubscript𝐸𝜆𝑛𝑥E_{\lambda}^{n}(x)\rightarrow\infty.

  3. 3.

    When 0<λ<1/e0𝜆1𝑒0<\lambda<1/e, the graph of Eλsubscript𝐸𝜆E_{\lambda} crosses the identity diagonal two times, at q𝑞q with Eλ(q)<1superscriptsubscript𝐸𝜆𝑞1E_{\lambda}^{\prime}(q)<1 and at p𝑝p with Eλ(p)>1superscriptsubscript𝐸𝜆𝑝1E_{\lambda}^{\prime}(p)>1, with which we see they are an attractor and a repellor, respectively.

Figure 1.1 presents some of these characteristics graphically, and the corresponding phase portraits are presented in figure 1.2. Figure 1.3 presents a common and very useful way of depicting bifurcations of real maps, which will be frequently used in this work: a bifurcation diagram; in it, the horizontal axis represents the values of the parameter λ𝜆\lambda in which the family is varied, and the vertical axis represents the values of the periodic points present for each fλsubscript𝑓𝜆f_{\lambda}.

Refer to caption
Figure 1.1: Graphs of Eλ(x)=λexsubscript𝐸𝜆𝑥𝜆superscript𝑒𝑥E_{\lambda}(x)=\lambda e^{x} where (a) λ>1/e𝜆1𝑒\lambda>1/e, (b) λ=1/e𝜆1𝑒\lambda=1/e, and (c) 0<λ<1/e0𝜆1𝑒0<\lambda<1/e. Reproduced from [Devaney(1989)].
Refer to caption
Figure 1.2: Phase portraits of Eλ(x)=λexsubscript𝐸𝜆𝑥𝜆superscript𝑒𝑥E_{\lambda}(x)=\lambda e^{x} for (a) λ>1/e𝜆1𝑒\lambda>1/e, (b) λ=1/e𝜆1𝑒\lambda=1/e, and (c) 0<λ<1/e0𝜆1𝑒0<\lambda<1/e. Reproduced from [Devaney(1989)].
Refer to caption
Figure 1.3: Bifurcation diagram of Eλ(x)=λexsubscript𝐸𝜆𝑥𝜆superscript𝑒𝑥E_{\lambda}(x)=\lambda e^{x}. x𝑥x is plotted against λ𝜆\lambda. Reproduced from [Devaney(1989)].

The above example suggests that bifurcations may occur when non-hyperbolic fixed points are present. Indeed, this the only case when bifurcations of fixed points occur (in the sense of one fixed point giving birth to two others), as the next theorem demonstrates.

Theorem 1.11.

Let fλsubscript𝑓𝜆f_{\lambda} be a one-parameter family of functions and suppose that fλ0(x0)=x0subscript𝑓subscript𝜆0subscript𝑥0subscript𝑥0f_{\lambda_{0}}(x_{0})=x_{0} and fλ0(x0)1subscriptsuperscript𝑓subscript𝜆0subscript𝑥01f^{\prime}_{\lambda_{0}}(x_{0})\neq 1. Then there are intervals I𝐼I about x0subscript𝑥0x_{0} and N𝑁N about λ0subscript𝜆0\lambda_{0} and a smooth function p:NI:𝑝𝑁𝐼p:N\rightarrow I such that p(λ0)=x0𝑝subscript𝜆0subscript𝑥0p(\lambda_{0})=x_{0} and fλ(p(λ))=p(λ)subscript𝑓𝜆𝑝𝜆𝑝𝜆f_{\lambda}(p(\lambda))=p(\lambda). Moreover, fλsubscript𝑓𝜆f_{\lambda} has no other fixed points in I𝐼I.

Proof.

Consider the function defined by G(λ,x)=fλ(x)x𝐺𝜆𝑥subscript𝑓𝜆𝑥𝑥G(\lambda,x)=f_{\lambda}(x)-x. By hypothesis, G(λ0,x0)=0𝐺subscript𝜆0subscript𝑥00G(\lambda_{0},x_{0})=0 and

Gx(λ0,x0)=fλ0(x0)10,𝐺𝑥subscript𝜆0subscript𝑥0subscriptsuperscript𝑓subscript𝜆0subscript𝑥010\frac{\partial G}{\partial x}(\lambda_{0},x_{0})=f^{\prime}_{\lambda_{0}}(x_{0})-1\neq 0, (1.1)

using the other hypothesis. Then, by the Implicit Function Theorem [Marsden(1974)], there are intervals I𝐼I about x0subscript𝑥0x_{0} and N𝑁N about λ0subscript𝜆0\lambda_{0}, and a smooth function p:NI:𝑝𝑁𝐼p:N\rightarrow I such that p(λ0)=x0𝑝subscript𝜆0subscript𝑥0p(\lambda_{0})=x_{0} and G(λ,p(λ))=0𝐺𝜆𝑝𝜆0G(\lambda,p(\lambda))=0 for all λN𝜆𝑁\lambda\in N; that is, p(λ)𝑝𝜆p(\lambda) is fixed by fλsubscript𝑓𝜆f_{\lambda} for λN𝜆𝑁\lambda\in N. Moreover, G(λ,x)0𝐺𝜆𝑥0G(\lambda,x)\neq 0 unless x=p(λ)𝑥𝑝𝜆x=p(\lambda). ∎

The above theorem obviously holds for periodic points by replacing fλsubscript𝑓𝜆f_{\lambda} with fλnsubscriptsuperscript𝑓𝑛𝜆f^{n}_{\lambda}.

1.7 Chaos

In this section we will precise what we understand by the mathematical concept of chaos. It must be said though, that there is not a single universally accepted definition for this concept, as variations occur; however, they preserve the same general “spirit”. We will here adopt one of the most popular definitions, due to [Devaney(1989)].

Before we can properly define the concept of chaos, we need some other auxiliary concepts.

Definition 1.11 (Topological transitivity).

Let (X,d)𝑋𝑑(X,\,d) be a metric space and f:XX:𝑓𝑋𝑋f:\,X\rightarrow X a map. Then f𝑓f is said to be topologically transitive if for any pair of nonempty open sets U,VX𝑈𝑉𝑋U,\,V\subset X there exists k>0𝑘0k>0 such that fk(U)Vsuperscript𝑓𝑘𝑈𝑉f^{k}(U)\cap V\neq\emptyset.

In plain words, what this definition states is that a map f𝑓f is topologically transitive if any arbitrarily small set of the domain of f𝑓f contains points that are mapped by f𝑓f to any other arbitrarily small set of the codomain in a finite number of iterations. An immediate consequence of topological transitivity is that the domain cannot be divided into two nonempty disjoint sets which are invariant under f𝑓f. It is easy to see that if f𝑓f has a dense orbit in X𝑋X then it is topologically transitive (the converse is also true for compact subsets of \mathbb{R} or S1superscript𝑆1S^{1}).

Another necessary concept for the definition of chaos is the following.

Definition 1.12 (Sensitive Dependence to Initial Conditions).

Let (X,d)𝑋𝑑(X,\,d) be a metric space and f:XX:𝑓𝑋𝑋f:\,X\rightarrow X a map. f𝑓f is said to posses sensitive dependence to initial conditions if there exists ε>0𝜀0\varepsilon>0 such that for any xX𝑥𝑋x\in X and any neighborhood N𝑁N of x𝑥x, there exists yN𝑦𝑁y\in N and n0𝑛0n\geq 0 such that |fn(x)fn(y)|>εsuperscript𝑓𝑛𝑥superscript𝑓𝑛𝑦𝜀|f^{n}(x)-f^{n}(y)|>\varepsilon.

Intuitively, sensitive dependence to initial conditions means that no matter how close two points are to each other, they will eventually be separated by at least some distance ε𝜀\varepsilon under a finite number of iterations of f𝑓f.

With these preliminary definitions we can now state the main definition of this section.

Definition 1.13.

Let (X,d)𝑋𝑑(X,\,d) be a metric space and f:XX:𝑓𝑋𝑋f:\,X\rightarrow X a map. Then f𝑓f is said to be chaotic on X𝑋X if

  1. 1.

    f𝑓f is topologically transitive,

  2. 2.

    the set of periodic points of f𝑓f is dense in X𝑋X, and

  3. 3.

    f𝑓f has sensitive dependence on initial conditions.

So a chaotic map “mixes up” its domain set by means of its topological transitivity but everywhere you look there are periodic points arbitrarily close and, finally, it is practically impossible to predict orbits numerically; these are the three elements that make up a chaotic map, under this definition. Actually, it was later shown [Banks et al.(1992)Banks, Brooks, Cairns, Davis, and Stacey] that topological transitivity and denseness of periodic points imply sensitive dependence to initial conditions, but no other pair of conditions imply the other. However, for continuous maps on intervals of \mathbb{R}, topological transitivity implies that the set of periodic points is dense [Vellekoop and Berglund(1994)], so that in this case, by the above, topological transitivity alone actually implies chaos.

1.8 The period doubling route to chaos

Perhaps one of the most important types of bifurcations is the period doubling bifurcation. This type of bifurcation will play a central role in the main results of this work. The most commonly know map which presents period doubling bifurcation is the famous logistic map. The issue is that the logistic quadratic map fλ(x)=λx(1x)subscript𝑓𝜆𝑥𝜆𝑥1𝑥f_{\lambda}(x)=\lambda x(1-x) has simple dynamics for 0λ30𝜆30\leq\lambda\leq 3 but it becomes chaotic when λ4𝜆4\lambda\geq 4 (we will precise the notion of chaos in the next section). The natural question here is: how exactly is it that fλsubscript𝑓𝜆f_{\lambda} becomes chaotic as λ𝜆\lambda increases? Where do the infinitely many periodic points come from when λ𝜆\lambda is large? In this section we will see that “period doubling” bifurcation is central to this.

Sarkovskii’s theorem provides a partial answer to the question of how many periodic points emerge when the parameter is varied. Before fλsubscript𝑓𝜆f_{\lambda} can have infinite periodic points of distinct periods, it must first have periodic points of all the periods of the form 2jsuperscript2𝑗2^{j} with j𝑗j\in\mathbb{N}. Local bifurcation theory provides two typical ways in which these points may emerge: by means of saddle-node bifurcations or through period doublings. The question is then which type of bifurcations occur as fλsubscript𝑓𝜆f_{\lambda} becomes chaotic.

The typical way in which fλsubscript𝑓𝜆f_{\lambda} becomes chaotic is that fλsubscript𝑓𝜆f_{\lambda} undergoes a series of period doubling bifurcations. This is not always the case, but it is a typical “route to chaos”.

Theorem 1.12 (Period Doubling Bifurcation).

Suppose that

  1. 1.

    fλ(x0)=x0subscript𝑓𝜆subscript𝑥0subscript𝑥0f_{\lambda}(x_{0})=x_{0} for all λ𝜆\lambda in an interval around λ0subscript𝜆0\lambda_{0}.

  2. 2.

    fλ(x0)=1superscriptsubscript𝑓𝜆subscript𝑥01f_{\lambda}^{\prime}(x_{0})=-1.

  3. 3.

    (fλ2)λ|λ=λ0(x0)0.evaluated-atsuperscriptsuperscriptsubscript𝑓𝜆2𝜆𝜆subscript𝜆0subscript𝑥00\frac{\partial(f_{\lambda}^{2})^{\prime}}{\partial\lambda}|_{\lambda=\lambda_{0}}(x_{0})\neq 0.

Then there exists an interval I𝐼I around x0subscript𝑥0x_{0} and a function p:I:𝑝𝐼p:I\rightarrow\mathbb{R} such that

fp(x)(x)xsubscript𝑓𝑝𝑥𝑥𝑥f_{p(x)}(x)\neq x

but

fp(x)2(x)=x.superscriptsubscript𝑓𝑝𝑥2𝑥𝑥f_{p(x)}^{2}(x)=x.
Proof.

Let G(λ,x)=fλ2(x)x𝐺𝜆𝑥subscriptsuperscript𝑓2𝜆𝑥𝑥G(\lambda,\,x)=f^{2}_{\lambda}(x)-x. Then, from hypotheses (1) and (2),

Gλ(λ0,x0)=0𝐺𝜆subscript𝜆0subscript𝑥00\frac{\partial G}{\partial\lambda}(\lambda_{0},\,x_{0})=0

so that we cannot apply the implicit function theorem directly. Thus, to rectify the situation, we set

B(λ,x)={G(λ,x)/(xx0)ifxx0,Gx(λ0,x0)ifx=x0.𝐵𝜆𝑥cases𝐺𝜆𝑥𝑥subscript𝑥0if𝑥subscript𝑥0𝐺𝑥subscript𝜆0subscript𝑥0if𝑥subscript𝑥0B(\lambda,\,x)=\begin{cases}G(\lambda,\,x)/(x-x_{0})&\mathrm{if}\,x\neq x_{0},\\ \frac{\partial G}{\partial x}(\lambda_{0},\,x_{0})&\mathrm{if}\,x=x_{0}.\end{cases} (1.2)

Straightforward calculation of the limits of the derivatives allow us to verify that B𝐵B is smooth and satisfies

Bx(λ0,x0)𝐵𝑥subscript𝜆0subscript𝑥0\displaystyle\frac{\partial B}{\partial x}(\lambda_{0},\,x_{0}) =122Gx2(λ0,x0)absent12superscript2𝐺superscript𝑥2subscript𝜆0subscript𝑥0\displaystyle=\frac{1}{2}\frac{\partial^{2}G}{\partial x^{2}}(\lambda_{0},x_{0}) (1.3)
2Bx2(λ0,x0)superscript2𝐵superscript𝑥2subscript𝜆0subscript𝑥0\displaystyle\frac{\partial^{2}B}{\partial x^{2}}(\lambda_{0},\,x_{0}) =133Gx3(λ0,x0).absent13superscript3𝐺superscript𝑥3subscript𝜆0subscript𝑥0\displaystyle=\frac{1}{3}\frac{\partial^{3}G}{\partial x^{3}}(\lambda_{0},x_{0}).

We now observe that, once again using hypotheses (1) and (2),

B(λ0,x0)𝐵subscript𝜆0subscript𝑥0\displaystyle B(\lambda_{0},\,x_{0}) =Gx(λ0,x0)absent𝐺𝑥subscript𝜆0subscript𝑥0\displaystyle=\frac{\partial G}{\partial x}(\lambda_{0},\,x_{0}) (1.4)
=xfλ02(x)|x=x01absentevaluated-at𝑥subscriptsuperscript𝑓2subscript𝜆0𝑥𝑥subscript𝑥01\displaystyle=\left.\frac{\partial}{\partial x}f^{2}_{\lambda_{0}}(x)\right|_{x=x_{0}}-1
=xfλ0(fλ0(x))|x=x01absentevaluated-at𝑥subscript𝑓subscript𝜆0subscript𝑓subscript𝜆0𝑥𝑥subscript𝑥01\displaystyle=\left.\frac{\partial}{\partial x}f_{\lambda_{0}}\left(f_{\lambda_{0}}(x)\right)\right|_{x=x_{0}}-1
=[yfλ0(y)|y=fλ0(x)xfλ0(x)]x=x01absentsubscriptdelimited-[]evaluated-at𝑦subscript𝑓subscript𝜆0𝑦𝑦subscript𝑓subscript𝜆0𝑥𝑥subscript𝑓subscript𝜆0𝑥𝑥subscript𝑥01\displaystyle=\left[\left.\frac{\partial}{\partial y}f_{\lambda_{0}}(y)\right|_{y=f_{\lambda_{0}}(x)}\cdot\frac{\partial}{\partial x}f_{\lambda_{0}}(x)\right]_{x=x_{0}}-1
=fλ0x(fλ0(x0))fλ0x(x0)1absentsubscript𝑓subscript𝜆0𝑥subscript𝑓subscript𝜆0subscript𝑥0subscript𝑓subscript𝜆0𝑥subscript𝑥01\displaystyle=\frac{\partial f_{\lambda_{0}}}{\partial x}(f_{\lambda_{0}}(x_{0}))\frac{\partial f_{\lambda_{0}}}{\partial x}(x_{0})-1
=[fλ0x(x0)]21absentsuperscriptdelimited-[]subscript𝑓subscript𝜆0𝑥subscript𝑥021\displaystyle=\left[\frac{\partial f_{\lambda_{0}}}{\partial x}(x_{0})\right]^{2}-1
=[fλ0(x0)]21absentsuperscriptdelimited-[]superscriptsubscript𝑓subscript𝜆0subscript𝑥021\displaystyle=\left[f_{\lambda_{0}}^{\prime}(x_{0})\right]^{2}-1
=(1)21absentsuperscript121\displaystyle=(-1)^{2}-1
=0.absent0\displaystyle=0.

Moreover, now using hypothesis (3),

Bλ(λ0,x0)𝐵𝜆subscript𝜆0subscript𝑥0\displaystyle\frac{\partial B}{\partial\lambda}(\lambda_{0},\,x_{0}) =λ(Gx(λ0,x0))absent𝜆𝐺𝑥subscript𝜆0subscript𝑥0\displaystyle=\frac{\partial}{\partial\lambda}\left(\frac{\partial G}{\partial x}(\lambda_{0},\,x_{0})\right) (1.5)
=λ(xfλ02(x0)1)absent𝜆𝑥subscriptsuperscript𝑓2subscript𝜆0subscript𝑥01\displaystyle=\frac{\partial}{\partial\lambda}\left(\frac{\partial}{\partial x}f^{2}_{\lambda_{0}}(x_{0})-1\right)
=2λxfλ02(x0)absentsuperscript2𝜆𝑥subscriptsuperscript𝑓2subscript𝜆0subscript𝑥0\displaystyle=\frac{\partial^{2}}{\partial\lambda\partial x}f^{2}_{\lambda_{0}}(x_{0})
=λ(fλ02)(x0)absent𝜆superscriptsubscriptsuperscript𝑓2subscript𝜆0subscript𝑥0\displaystyle=\frac{\partial}{\partial\lambda}(f^{2}_{\lambda_{0}})^{\prime}(x_{0})
0.absent0\displaystyle\neq 0.

Therefore, by the implicit function theorem [Marsden(1974)], there exists a C1superscript𝐶1C^{1} map p(x)𝑝𝑥p(x) defined on an interval around x0subscript𝑥0x_{0} such that p(x0)=λ0𝑝subscript𝑥0subscript𝜆0p(x_{0})=\lambda_{0} and B(p(x),x)=0𝐵𝑝𝑥𝑥0B(p(x),x)=0. So that, for xx0𝑥subscript𝑥0x\neq x_{0},

1xx0G(p(x),x)=0.1𝑥subscript𝑥0𝐺𝑝𝑥𝑥0\frac{1}{x-x_{0}}G(p(x),x)=0.

Consequently, G(p(x),x)=0𝐺𝑝𝑥𝑥0G(p(x),x)=0 and then fp(x)2(x)=xsubscriptsuperscript𝑓2𝑝𝑥𝑥𝑥f^{2}_{p(x)}(x)=x and thus x𝑥x is of period 2 for λ=p(x)𝜆𝑝𝑥\lambda=p(x) and for x𝑥x in an interval around x0subscript𝑥0x_{0}. Notice that x𝑥x is not fixed by fp(x)subscript𝑓𝑝𝑥f_{p(x)} because we can apply theorem 1.11 in this case and, then, the curve of fixed points is unique.

On the other hand, from the chain rule we have

ddxB(p(x),x)=Bx+Bλp(x)=0.𝑑𝑑𝑥𝐵𝑝𝑥𝑥𝐵𝑥𝐵𝜆superscript𝑝𝑥0\frac{d}{dx}B(p(x),x)=\frac{\partial B}{\partial x}+\frac{\partial B}{\partial\lambda}p^{\prime}(x)=0.

From which,

p(x)=Bx(p(x),x)Bλ(p(x),x.p^{\prime}(x)=-\frac{\frac{\partial B}{\partial x}(p(x),x)}{\frac{\partial B}{\partial\lambda}(p(x),x}. (1.6)

We know that, for x=x0𝑥subscript𝑥0x=x_{0}, the denominator in (1.6) is different from zero and, on the other hand, the numerator is equal to 122Gx212superscript2𝐺superscript𝑥2\frac{1}{2}\frac{\partial^{2}G}{\partial x^{2}} and

2Gx2(λ0,x0)superscript2𝐺superscript𝑥2subscript𝜆0subscript𝑥0\displaystyle\frac{\partial^{2}G}{\partial x^{2}}(\lambda_{0},x_{0}) =12{fλ′′(fλ(x))[fλ(x)]2+fλ(fλ(x))fλ′′(x)}|x=x0,λ=λ0absentevaluated-at12subscriptsuperscript𝑓′′𝜆subscript𝑓𝜆𝑥superscriptdelimited-[]subscriptsuperscript𝑓𝜆𝑥2subscriptsuperscript𝑓𝜆subscript𝑓𝜆𝑥subscriptsuperscript𝑓′′𝜆𝑥formulae-sequence𝑥subscript𝑥0𝜆subscript𝜆0\displaystyle=\frac{1}{2}\left.\left\{f^{\prime\prime}_{\lambda}(f_{\lambda}(x))[f^{\prime}_{\lambda}(x)]^{2}+f^{\prime}_{\lambda}(f_{\lambda}(x))f^{\prime\prime}_{\lambda}(x)\right\}\right|_{x=x_{0},\,\lambda=\lambda_{0}} (1.7)
=12{fλ0′′(x0)[1]2fλ0′′(x0)}absent12subscriptsuperscript𝑓′′subscript𝜆0subscript𝑥0superscriptdelimited-[]12subscriptsuperscript𝑓′′subscript𝜆0subscript𝑥0\displaystyle=\frac{1}{2}\left\{f^{\prime\prime}_{\lambda_{0}}(x_{0})[-1]^{2}-f^{\prime\prime}_{\lambda_{0}}(x_{0})\right\}
=0.absent0\displaystyle=0.

Where we have once again used hypothesis (2). Ergo, p(x0)=0superscript𝑝subscript𝑥00p^{\prime}(x_{0})=0. ∎

Also, differentiating equation (1.6) again and evaluating in x=x0𝑥subscript𝑥0x=x_{0}, it is easy to see that

p′′(x0)=2Bx2(λ0,x0)Bλ(λ0,x0).superscript𝑝′′subscript𝑥0superscript2𝐵superscript𝑥2subscript𝜆0subscript𝑥0𝐵𝜆subscript𝜆0subscript𝑥0p^{\prime\prime}(x_{0})=-\frac{\frac{\partial^{2}B}{\partial x^{2}}(\lambda_{0},x_{0})}{\frac{\partial B}{\partial\lambda}(\lambda_{0},x_{0})}. (1.8)

We already know that the denominator in (1.8) above is not null and, also,

2Bx2(λ0,x0)superscript2𝐵superscript𝑥2subscript𝜆0subscript𝑥0\displaystyle\frac{\partial^{2}B}{\partial x^{2}}(\lambda_{0},x_{0}) =133Gx3absent13superscript3𝐺superscript𝑥3\displaystyle=\frac{1}{3}\frac{\partial^{3}G}{\partial x^{3}} (1.9)
=13{fλ′′′(fλ(x))[fλ(x)]3+2fλ′′(fλ(x))fλ(x)+fλ′′(fλ(x))fλ′′(x)fλ(x)\displaystyle=\frac{1}{3}\left\{f^{\prime\prime\prime}_{\lambda}(f_{\lambda}(x))\left[f^{\prime}_{\lambda}(x)\right]^{3}+2f^{\prime\prime}_{\lambda}(f_{\lambda}(x))f^{\prime}_{\lambda}(x)+f^{\prime\prime}_{\lambda}(f_{\lambda}(x))f^{\prime\prime}_{\lambda}(x)f^{\prime}_{\lambda}(x)\right.
+fλ(fλ(x))fλ′′′(x)}|x=x0,λ=λ0\displaystyle\quad\left.\left.+f^{\prime}_{\lambda}(f_{\lambda}(x))f^{\prime\prime\prime}_{\lambda}(x)\right\}\right|_{x=x_{0},\,\lambda=\lambda_{0}}
=13{2fλ0′′′(x0)3[fλ0′′(x0)]2}absent132subscriptsuperscript𝑓′′′subscript𝜆0subscript𝑥03superscriptdelimited-[]subscriptsuperscript𝑓′′subscript𝜆0subscript𝑥02\displaystyle=\frac{1}{3}\left\{-2f^{\prime\prime\prime}_{\lambda_{0}}(x_{0})-3[f^{\prime\prime}_{\lambda_{0}}(x_{0})]^{2}\right\}
=23{fλ0′′′(x0)fλ0(x0)32[fλ0′′(x0)fλ0(x0)]2}absent23subscriptsuperscript𝑓′′′subscript𝜆0subscript𝑥0subscriptsuperscript𝑓subscript𝜆0subscript𝑥032superscriptdelimited-[]subscriptsuperscript𝑓′′subscript𝜆0subscript𝑥0subscriptsuperscript𝑓subscript𝜆0subscript𝑥02\displaystyle=\frac{2}{3}\left\{\frac{f^{\prime\prime\prime}_{\lambda_{0}}(x_{0})}{f^{\prime}_{\lambda_{0}}(x_{0})}-\frac{3}{2}\left[\frac{f^{\prime\prime}_{\lambda_{0}}(x_{0})}{f^{\prime}_{\lambda_{0}}(x_{0})}\right]^{2}\right\}
=23Sfλ0(x0),absent23𝑆subscript𝑓subscript𝜆0subscript𝑥0\displaystyle=\frac{2}{3}\,Sf_{\lambda_{0}}(x_{0}),

where Sfλ𝑆subscript𝑓𝜆Sf_{\lambda} is the Schwarzian derivative of fλsubscript𝑓𝜆f_{\lambda} and, yet once again, we used hypothesis (2) from the theorem above. Therefore, substituting in equation (1.8) we have

p′′(x0)=23Sfλ0(x0)λ(fλ02)(x0).superscript𝑝′′subscript𝑥023𝑆subscript𝑓subscript𝜆0subscript𝑥0𝜆superscriptsubscriptsuperscript𝑓2subscript𝜆0subscript𝑥0p^{\prime\prime}(x_{0})=-\frac{-\frac{2}{3}\,Sf_{\lambda_{0}}(x_{0})}{\frac{\partial}{\partial\lambda}(f^{2}_{\lambda_{0}})^{\prime}(x_{0})}. (1.10)

So that, if Sfλ0𝑆subscript𝑓𝜆0Sf_{\lambda}\neq 0 then p′′(x0)0superscript𝑝′′subscript𝑥00p^{\prime\prime}(x_{0})\neq 0 and the curve p(x)𝑝𝑥p(x) is either concave left or right. Along with p(x0)=0𝑝subscript𝑥00p(x_{0})=0 this gives us an idea of the geometry of the period 2 curve p𝑝p near the bifurcation point x0subscript𝑥0x_{0}. Notice that x0subscript𝑥0x_{0} does not bifurcate in the sense of giving birth to another fixed point (as was the case in the pitchfork bifurcation) but, rather, x0subscript𝑥0x_{0} continues to be the only fixed point in a interval around λ0subscript𝜆0\lambda_{0} (see theorem 1.11).

In the case of the logistic map, the period doubling bifurcation repeats over and over again for the fixed point x=λ1λsuperscript𝑥𝜆1𝜆x^{*}=\frac{\lambda-1}{\lambda} as the value of the parameter λ𝜆\lambda is increased, starting from λ=3𝜆3\lambda=3:

  • Between λ0=1<λ<3=λ1subscript𝜆01𝜆3subscript𝜆1\lambda_{0}=1<\lambda<3=\lambda_{1}, fλsubscript𝑓𝜆f_{\lambda} has a single stable fixed point, xsuperscript𝑥x^{*} (of period 20=1superscript2012^{0}=1).

  • For λ1=3<λ<1+6=λ2subscript𝜆13𝜆16subscript𝜆2\lambda_{1}=3<\lambda<1+\sqrt{6}=\lambda_{2}, xsuperscript𝑥x^{*} loses its stability and ’gives rise’ to an attracting cycle of period two (21superscript212^{1}), whose orbital points separate from xsuperscript𝑥x^{*} as λ𝜆\lambda increases.

  • For λ2=1+6<λ<3.54409=λ3subscript𝜆216𝜆3.54409subscript𝜆3\lambda_{2}=1+\sqrt{6}<\lambda<3.54409=\lambda_{3} [Elaydi(2000)], the attracting cycle of period two loses in turn its stability and gives rise to an attracting cycle of period 22superscript222^{2}.

  • For λ>3.54409=λ3𝜆3.54409subscript𝜆3\lambda>3.54409=\lambda_{3}, the period four cycle also loses its stability and bifurcates into an asymptotically stable cycle of period 23superscript232^{3}.

The above mentioned process repeats itself indefinitely and produces a sequence {λk}k=1superscriptsubscriptsubscript𝜆𝑘𝑘1\{\lambda_{k}\}_{k=1}^{\infty}. The Feigenbaum sequence {λk}k=0superscriptsubscriptsubscript𝜆𝑘𝑘0\{\lambda_{k}\}_{k=0}^{\infty} describes the values of the parameter λ𝜆\lambda for which period doubling bifurcations arise in a unimodal map [Feigenbaum(1978)]. The sequence converges exponentially. The important point to remark here is that the rate of convergence of these sequences (whose values depend on the specific unimodal map), is constant for every unimodal function and is given by δ4.669201609𝛿4.669201609\delta\approx 4.669201609\cdots, which gives rise to the so-called property of universality. In fact, its importance deserved naming it Feigenbaum’s constant, after [Feigenbaum(1978)] discovered it in [Feigenbaum(1978)] (though amazingly ignored for over a decade).

1.9 Topological conjugacy

Here we will define the important concept of topological conjugacy which relates the dynamics of two different maps by means of a third one. The importance of conjugacy is that it preserves the stability and chaotic properties of a map, so that one map can be analyzed by means of analyzing another one, presumably simpler, so that the analysis becomes easier.

Definition 1.14 (Topological Conjugacy).

Let f:AA:𝑓𝐴𝐴f:A\rightarrow A and g:BB:𝑔𝐵𝐵g:B\rightarrow B be two maps. Then f𝑓f and g𝑔g are said to be topologically conjugate, denoted fg𝑓𝑔f\approx g, if there exists a homeomorphism h:AB:𝐴𝐵h:A\rightarrow B such that hg=gh𝑔𝑔h\circ g=g\circ h. The homeomorphism is called a topological conjugacy. We also say that f𝑓f is hh-conjugate to g𝑔g to emphasize the importance of the homeomorphism hh.

It is easy to prove that conjugacy is an equivalence relation. Topologically conjugate maps have equivalent dynamics. For example, if fg𝑓𝑔f\approx g and x𝑥x is a fixed point of f𝑓f then h(x)𝑥h(x) is a fixed point of g𝑔g; and, also, fkgksuperscript𝑓𝑘superscript𝑔𝑘f^{k}\approx g^{k} for all k𝑘k\in\mathbb{N}. Similarly, periodic orbits of f𝑓f are mapped to analogue periodic orbits of g𝑔g.

Theorem 1.13.

Let f:AA:𝑓𝐴𝐴f:A\rightarrow A be hh-conjugate to g:BB:𝑔𝐵𝐵g:B\rightarrow B. If f𝑓f is chaotic on A𝐴A, then g𝑔g is chaotic on B𝐵B.

Theorem 1.13 will play a key role in the importance of the study of “canonical polynomial maps” in chapters 3, 4 and 5. The proof of this theorem can be found in [Elaydi(2000)], where it can be seen that, in fact, the topological conjugacy hh does not need to be a homeomorphism for theorem 1.13 to hold, it only needs to be onto, continuous and open, which leads to the concept of

Definition 1.15 (Semiconjugacy).

The maps f:AA:𝑓𝐴𝐴f:A\rightarrow A and g:BB:𝑔𝐵𝐵g:B\rightarrow B are said to be semiconjugate if there exists a map h:AB:𝐴𝐵h:A\rightarrow B onto, continuous and open, such that hf=gh𝑓𝑔h\circ f=g\circ h.

That been stated, it can be shown that theorem 1.13 holds with “conjugate” replaced by “semiconjugate”.

Chapter 2 Regular-Reversal quadratic maps

In this chapter the work of [Solís and Jódar(2004)] will be discussed, as the main precedent to the present work, where an application of bifurcation theory and chaos in discrete dynamical systems is analyzed to determine the conditions in which chaos arises for a family of unimodal maps, namely quadratic maps. If the asymptotic behavior and parametric dependency of quadratic maps is understood, the results can be generalized to more complex maps, starting to set a precedent for cubic maps and polynomial maps of higher order. The only original contribution in this chapter is given by some of the examples in the corresponding section below.

The main objective of this chapter is to analyze, for a given unimodal quadratic map, under what circumstances does the property of period doubling hold and when it disappears, which means that chaos is no longer a possibility, so as to be able to control the appearance (or not) of chaotic behavior by controlling the value of the parameter.

2.1 Unimodal maps

Consider a one-parameter family of real discrete dynamical systems of the form xn+1=g(xn,λ)subscript𝑥𝑛1𝑔subscript𝑥𝑛𝜆x_{n+1}=g(x_{n},\lambda) with iteration function g(xn,λ)=gλ(xn)𝑔subscript𝑥𝑛𝜆subscript𝑔𝜆subscript𝑥𝑛g(x_{n},\lambda)=g_{\lambda}(x_{n}), g:J×I:𝑔𝐽𝐼g:J\times I\rightarrow\mathbb{R}, I𝐼I, J𝐽J intervals, and g𝑔g smooth on x𝑥x and λ𝜆\lambda.

Definition 2.1 (Unimodal Map).

An iteration function (map) f:II:𝑓𝐼𝐼f:I\subset\mathbb{R}\rightarrow I, with I𝐼I an interval, is called unimodal if it is smooth111This requirement follows from the computability of the Schwarzian derivative. (at least C3(I)superscript𝐶3𝐼C^{3}(I)) and with a unique maximal point. By extension, we call a family of maps unimodal if every member of the family is unimodal.

In the following, let g(xn,λ)𝑔subscript𝑥𝑛𝜆g(x_{n},\lambda) be unimodal and with a fixed point x=xp(λ)𝑥subscript𝑥𝑝𝜆x=x_{p}(\lambda).

Definition 2.2 (Eigenvalue function).

The eigenvalue function ϕ::italic-ϕ\phi:\mathbb{R}\rightarrow\mathbb{R} corresponding to the fixed point x=xp(λ)𝑥subscript𝑥𝑝𝜆x=x_{p}(\lambda) of the one-parameter family of maps g(xn,λ)𝑔subscript𝑥𝑛𝜆g(x_{n},\lambda) is

ϕ(λ)=gx(xp(λ)),italic-ϕ𝜆𝑔𝑥subscript𝑥𝑝𝜆\phi(\lambda)=\frac{\partial g}{\partial x}(x_{p}(\lambda)), (2.1)

defined for the values of λ𝜆\lambda for which the fixed point exists.

That is, the eigenvalue function gives us the multiplier of a fixed point as a function of the parameter of the family of maps. The eigenvalue function for fixed λ𝜆\lambda is simply the derivative of the map at the fixed point, i.e. the “traditional” multiplier. Notice that ϕitalic-ϕ\phi is continuous since g𝑔g is smooth. The fixed point xp(λ)subscript𝑥𝑝𝜆x_{p}(\lambda) is asymptotically stable for the values of λ𝜆\lambda for which 1<ϕ(λ)<11italic-ϕ𝜆1-1<\phi(\lambda)<1, agreeing with our notion of a hyperbolic attracting fixed point.

Definition 2.3 (Region of Type 1).

The region of type 1 of xp(λ)subscript𝑥𝑝𝜆x_{p}(\lambda) is the set {(λ,ϕ(λ))|1<ϕ(λ)<1}2conditional-set𝜆italic-ϕ𝜆1italic-ϕ𝜆1superscript2\{(\lambda,\,\phi(\lambda))|-1<\phi(\lambda)<1\}\subset\mathbb{R}^{2}.

The region of type 1 of xpsubscript𝑥𝑝x_{p} is the set of values of the parameter λ𝜆\lambda for which xpsubscript𝑥𝑝x_{p} is a stable fixed point, along with its corresponding values of the eigenvalue function ϕitalic-ϕ\phi (multipliers). Therefore, the region of type 1 of a fixed point is its stability region as a function of the parameter λ𝜆\lambda. Since g(x,λ)𝑔𝑥𝜆g(x,\lambda) is smooth on both λ𝜆\lambda and x𝑥x, regions of type 1 consist of the countable union of connected subsets of 2superscript2\mathbb{R}^{2}. In general, we can define regions of higher type respect to the fixed points of the k𝑘k-th iteration of g𝑔g, i.e. the periodic points of prime period k𝑘k.

Definition 2.4 (Region of Type k𝑘k).

If the family of maps xn+1=g(xn,λ)subscript𝑥𝑛1𝑔subscript𝑥𝑛𝜆x_{n+1}=g(x_{n},\lambda), not necessarily unimodal, has an isolated periodic point of prime period k1,𝑘1k\geq 1,, xpk(λ)superscriptsubscript𝑥𝑝𝑘𝜆x_{p}^{k}(\lambda), then we can define the k𝑘k-th eigenvalue function ϕk::subscriptitalic-ϕ𝑘\phi_{k}:\mathbb{R}\rightarrow\mathbb{R} as

ϕk(λ)=g(k)x(xpk(λ)).subscriptitalic-ϕ𝑘𝜆superscript𝑔𝑘𝑥superscriptsubscript𝑥𝑝𝑘𝜆\phi_{k}(\lambda)=\frac{\partial g^{(k)}}{\partial x}(x_{p}^{k}(\lambda)). (2.2)

Notice that g(k)xsuperscript𝑔𝑘𝑥\frac{\partial g^{(k)}}{\partial x} refers to the first derivative of the k𝑘k-th iteration of g𝑔g and not the k𝑘k-th derivative of g𝑔g, as in standard calculus notation. The set {(λ,ϕk(λ))2|xpk(λ) is stable}conditional-set𝜆subscriptitalic-ϕ𝑘𝜆superscript2superscriptsubscript𝑥𝑝𝑘𝜆 is stable\{(\lambda,\phi_{k}(\lambda))\in\mathbb{R}^{2}|x_{p}^{k}(\lambda)\text{ is stable}\} is denoted as the region of type k𝑘k for xpksuperscriptsubscript𝑥𝑝𝑘x_{p}^{k}.

The region of type k𝑘k of the periodic point xpksuperscriptsubscript𝑥𝑝𝑘x_{p}^{k} is simply the region of type 1 of the function g(k)superscript𝑔𝑘g^{(k)}, of which xpksuperscriptsubscript𝑥𝑝𝑘x_{p}^{k} is a fixed point. Moreover, by the chain rule,

ϕk(λ)=g(k)x=gλ(xpk)gλ(gλ(xpk))gλ(gλ(2)(xpk))gλ(gλ(k1)(xpk))subscriptitalic-ϕ𝑘𝜆superscript𝑔𝑘𝑥subscriptsuperscript𝑔𝜆superscriptsubscript𝑥𝑝𝑘subscriptsuperscript𝑔𝜆subscript𝑔𝜆superscriptsubscript𝑥𝑝𝑘subscriptsuperscript𝑔𝜆superscriptsubscript𝑔𝜆2superscriptsubscript𝑥𝑝𝑘subscriptsuperscript𝑔𝜆superscriptsubscript𝑔𝜆𝑘1superscriptsubscript𝑥𝑝𝑘\phi_{k}(\lambda)=\frac{\partial g^{(k)}}{\partial x}=g^{\prime}_{\lambda}(x_{p}^{k})g^{\prime}_{\lambda}(g_{\lambda}(x_{p}^{k}))g^{\prime}_{\lambda}(g_{\lambda}^{(2)}(x_{p}^{k}))\cdots g^{\prime}_{\lambda}(g_{\lambda}^{(k-1)}(x_{p}^{k}))

so that it suffices that the absolute value of the product of the multipliers along the orbit of a periodic point xpksuperscriptsubscript𝑥𝑝𝑘x_{p}^{k} is less than one in order to guarantee the stability of xpksuperscriptsubscript𝑥𝑝𝑘x_{p}^{k}. Nonetheless, the existence of periodic points of period k>1𝑘1k>1 is not assured even for a unimodal map in general. Thereafter, the periodic point, x=xpk(λ)𝑥superscriptsubscript𝑥𝑝𝑘𝜆x=x_{p}^{k}(\lambda), when it exists, is stable in an interval of the parameter λ𝜆\lambda with ends given by ϕk1(1)superscriptsubscriptitalic-ϕ𝑘11\phi_{k}^{-1}(1) and ϕk(1)(1)superscriptsubscriptitalic-ϕ𝑘11\phi_{k}^{(-1)}(-1). We suppose here that the function ϕk1superscriptsubscriptitalic-ϕ𝑘1\phi_{k}^{-1} exists, at least locally, i.e. ϕk(λ)0subscriptsuperscriptitalic-ϕ𝑘𝜆0\phi^{\prime}_{k}(\lambda)\neq 0 for all λ𝜆\lambda in an interval around a value of interest λ0subscript𝜆0\lambda_{0} associated with the periodic point xpk(λ0)=x0ksuperscriptsubscript𝑥𝑝𝑘subscript𝜆0superscriptsubscript𝑥0𝑘x_{p}^{k}(\lambda_{0})=x_{0}^{k}, according to the Inverse Function Theorem [Protter(1998)]. A simpler way of saying the latter is that xpksuperscriptsubscript𝑥𝑝𝑘x_{p}^{k} will be attracting as long as 1<ϕk(xpk)<11subscriptitalic-ϕ𝑘superscriptsubscript𝑥𝑝𝑘1-1<\phi_{k}(x_{p}^{k})<1, but notice that depending on the form of the graph of ϕksubscriptitalic-ϕ𝑘\phi_{k}, the eigenvalue function may never leave this region therefore undefining ϕk1superscriptsubscriptitalic-ϕ𝑘1\phi_{k}^{-1} for λ=±1𝜆plus-or-minus1\lambda=\pm 1. Another way of looking at the region of k𝑘k is that the type k𝑘k of a given region is given by the order k𝑘k of the attracting k𝑘k-cycle existing in that region.

Now, since a point (λ,ϕ(λ))2𝜆italic-ϕ𝜆superscript2(\lambda,\phi(\lambda))\in\mathbb{R}^{2} can only belong to a unique and specific region of a given periodic point, we can state the following definition.

Definition 2.5 (Regular and Reversal Maps).

Let 𝒜𝒜\mathcal{A}\subset\mathbb{R} be an interval. A map xn+1=g(xn,λ)subscript𝑥𝑛1𝑔subscript𝑥𝑛𝜆x_{n+1}=g(x_{n},\lambda) is called a regular map (respectively, reversal) in 𝒜𝒜\mathcal{A} if it has an isolated periodic point such that its associated eigenvalue function ϕitalic-ϕ\phi has the property that if λ1,λ2𝒜subscript𝜆1subscript𝜆2𝒜\lambda_{1},\,\lambda_{2}\in\mathcal{A}, with λ1<λ2subscript𝜆1subscript𝜆2\lambda_{1}<\lambda_{2}, then the type of the region containing the point (λ2,ϕ(λ2))subscript𝜆2italic-ϕsubscript𝜆2(\lambda_{2},\phi(\lambda_{2})) is higher (respectively, lower) or equal than the type of the region containing the point (λ1,ϕ(λ1))subscript𝜆1italic-ϕsubscript𝜆1(\lambda_{1},\phi(\lambda_{1})).

In a regular map, as the value of λ𝜆\lambda increases, new attracting periodic points of progressively higher periods are created and, therefore, the type of the regions associated with these points equally rises (or at least does not decrease) progressively. On the other hand, in a reversal map, as the value of the parameter λ𝜆\lambda increases, higher order attracting periodic points disappear and only progressively lower order attracting periodic points remain, so that the type of the associated regions decreases correspondingly.

Example 2.1 (The Logistic Map).

The logistic map fλ(x)=λx(1x)subscript𝑓𝜆𝑥𝜆𝑥1𝑥f_{\lambda}(x)=\lambda x(1-x) is a classical example of a regular map, since it satisfies the definition 2.5 in the interval [0,λ]0subscript𝜆[0,\lambda_{\infty}], with λ3.570subscript𝜆3.570\lambda_{\infty}\approx 3.570. Its eigenvalue function is

ϕ(λ)=λ(12λx)|x=xp(λ),italic-ϕ𝜆evaluated-at𝜆12𝜆𝑥𝑥subscript𝑥𝑝𝜆\phi(\lambda)=\left.\lambda(1-2\lambda x)\right|_{x=x_{p}(\lambda)},

for the nonzero fixed point xp(λ)subscript𝑥𝑝𝜆x_{p}(\lambda). In table 2.1 some type k𝑘k regions are shown for the logistic map.

Type of the region (k𝑘k) Interval [(λk1,λk)subscript𝜆𝑘1subscript𝜆𝑘(\lambda_{k-1},\lambda_{k})]
1 (0, 3)
2 (3, 3.449489…)
3 (3.449489…, 3.544090…)
4 (3.544090…, 3.564407…)
5 (3.564407…, 3.568759…)
6 (3.568759…, 3.569692…)
7 (3.569692…, 3.569891…)
Table 2.1: Table of regions of type k𝑘k, k=1,2,,7𝑘127k=1,2,...,7, for the logistic map. The logistic map is a regular map (see definition 2.5). Data reproduced from [Elaydi(2000), Table 1.1, p.41]
Example 2.2 (The Exponential Map).

Consider the one-parameter family of maps Eλ(x)=λexsubscript𝐸𝜆𝑥𝜆superscript𝑒𝑥E_{\lambda}(x)=\lambda e^{x}, with λ<0𝜆0\lambda<0. In figure 2.1 are shown three important cases of the graphs of the familyEλsubscript𝐸𝜆E_{\lambda}. When λ<e𝜆𝑒\lambda<-e the map has a repelling fixed point because of theorem 1.1 (see figure 2.1c); when λ=e𝜆𝑒\lambda=-e, Eλ(1)=1subscript𝐸𝜆11E_{\lambda}(-1)=-1 and Eλ(1)=11subscriptsuperscript𝐸𝜆111E^{\prime}_{\lambda}(-1)=-1\neq 1, so that -1 is a non-hyperbolic fixed point and, by theorem 1.11, there is an interval around λ=e𝜆𝑒\lambda=-e in which this fixed point exists and is unique; and when λ>e𝜆𝑒\lambda>-e the fixed point is attracting also because of theorem 1.1. Therefore the fixed point of Eλsubscript𝐸𝜆E_{\lambda} undergoes a change in stability from unstable to stable as the value of λ𝜆\lambda increases from λ<e𝜆𝑒\lambda<-e to λ>e𝜆𝑒\lambda>e, with the bifurcation value being exactly λ=e𝜆𝑒\lambda=-e. Also, it is easy to see that Eλ2subscriptsuperscript𝐸2𝜆E^{2}_{\lambda} is an increasing function that is concave upward if Eλ(x)<1subscript𝐸𝜆𝑥1E_{\lambda}(x)<-1 and concave downward if Eλ(x)>1subscript𝐸𝜆𝑥1E_{\lambda}(x)>-1 (see figure 2.2). Thus Eλ2subscriptsuperscript𝐸2𝜆E^{2}_{\lambda} has two fixed points at the points q1subscript𝑞1q_{1} and q2subscript𝑞2q_{2} shown in figure 2.2 when λ<e𝜆𝑒\lambda<-e and they disappear when λ𝜆\lambda increases to e𝑒-e. Since we know that Eλsubscript𝐸𝜆E_{\lambda} has a single fixed point, these must in fact be periodic points of period 2. Note that, as these period 2 points lose their “attractiveness”, the fixed point acquires it, so that there is an “exchange of stability”. Also note that, at λ=e𝜆𝑒\lambda=-e, λ(Eλ2)(1)0𝜆superscriptsubscriptsuperscript𝐸2𝜆10\frac{\partial}{\partial\lambda}\left(E^{2}_{\lambda}\right)^{\prime}(-1)\neq 0, so that we can apply theorem 1.12 formally prove that a period doubling bifurcation effectively takes place. The bifurcation diagram of this example is shown in figure 2.3, where it is easy to see that it satisfies the definition of a reversal map.

Refer to caption
Figure 2.1: The graphs of the one-parameter family of maps Eλ(x)=λexsubscript𝐸𝜆𝑥𝜆superscript𝑒𝑥E_{\lambda}(x)=\lambda e^{x} for (a) e<λ<0𝑒𝜆0-e<\lambda<0, (b) λ=e𝜆𝑒\lambda=-e, and (c) λ<e𝜆𝑒\lambda<-e. Reproduced from [Devaney(1989)].
Refer to caption
Figure 2.2: The graphs of Eλ2(x)subscriptsuperscript𝐸2𝜆𝑥E^{2}_{\lambda}(x) for (a) e<λ<0𝑒𝜆0-e<\lambda<0, (b) λ=e𝜆𝑒\lambda=-e, and (c) λ<e𝜆𝑒\lambda<-e. Reproduced from [Devaney(1989)].
Refer to caption
Figure 2.3: The bifurcation diagram of Eλ(x)=λexsubscript𝐸𝜆𝑥𝜆superscript𝑒𝑥E_{\lambda}(x)=\lambda e^{x}. Notice that the “vertical” curve is the curve of period 2 attracting points. Reproduced from [Devaney(1989)].
Definition 2.6 (Regular-Reversal Map).

Let 𝒜𝒜\mathcal{A} be a nonempty interval in \mathbb{R}. A map xn+1=g(xn,λ)subscript𝑥𝑛1𝑔subscript𝑥𝑛𝜆x_{n+1}=g(x_{n},\lambda) is called a regular-reversal map in 𝒜𝒜\mathcal{A} if the interval 𝒜𝒜\mathcal{A} can be decomposed in two nonempty disjoint intervals A1subscript𝐴1A_{1} and A2subscript𝐴2A_{2} such that the map is regular in A1subscript𝐴1A_{1} and reversal in A2subscript𝐴2A_{2}.

In a regular-reversal map, there is a value of the parameter λ𝜆\lambda, say λ0subscript𝜆0\lambda_{0}, that divides the interval of the family of maps xn+1=g(xn,λ)subscript𝑥𝑛1𝑔subscript𝑥𝑛𝜆x_{n+1}=g(x_{n},\lambda) in two, in one of which it is regular and another one in which it is reversal. A regular-reversal map is interesting because it shows exactly the parametric range where periodicity takes place. So far, it has been difficult to determine for which maps this phenomenon occurs. It is for this reason that precedents have focused only in quadratic polynomial maps [Solís and Jódar(2004)], but also because quadratic polynomials give a complete family of unimodal maps with negative Schwarzian derivative and they represent the simplest example of unimodal maps. In this work, we will focus in higher order polynomial maps, particularly cubic, and we will be able to determine a simple way to create regular-reversal polynomial maps of any degrees. We can also define reversal-regular maps in a similar way.

2.2 Quadratic maps

Consider the family of quadratic discrete dynamical systems given by

yn+1=α(λ)yn2+(β(λ)+1)yn+γ(λ),subscript𝑦𝑛1𝛼𝜆superscriptsubscript𝑦𝑛2𝛽𝜆1subscript𝑦𝑛𝛾𝜆y_{n+1}=\alpha(\lambda)y_{n}^{2}+(\beta(\lambda)+1)y_{n}+\gamma(\lambda), (2.3)

where α𝛼\alpha, β𝛽\beta and γ𝛾\gamma are smooth functions of the parameter λ𝜆\lambda and α(λ)0𝛼𝜆0\alpha(\lambda)\neq 0 for all λ𝜆\lambda.

The fixed points of this system are obtained by solving the general quadratic equation

α(λ)y2+β(λ)y+γ(λ)=0𝛼𝜆superscript𝑦2𝛽𝜆𝑦𝛾𝜆0\alpha(\lambda)y^{2}+\beta(\lambda)y+\gamma(\lambda)=0

and they are given by the general quadratic formula

y0=(2α)1(β+β24αγ),y1=(2α)1(ββ24αγ).formulae-sequencesubscript𝑦0superscript2𝛼1𝛽superscript𝛽24𝛼𝛾subscript𝑦1superscript2𝛼1𝛽superscript𝛽24𝛼𝛾y_{0}=(2\alpha)^{-1}\left(-\beta+\sqrt{\beta^{2}-4\alpha\gamma}\right),\quad y_{1}=(2\alpha)^{-1}\left(-\beta-\sqrt{\beta^{2}-4\alpha\gamma}\right).

We will assume that β24αγ>0superscript𝛽24𝛼𝛾0\beta^{2}-4\alpha\gamma>0 in order to have two distinct real fixed points. We can simplify the system introducing the change of variable

xn=Ayn+Bsubscript𝑥𝑛𝐴subscript𝑦𝑛𝐵x_{n}=Ay_{n}+B

where A=α1𝐴superscript𝛼1A=\alpha^{-1} and B=(2α)1(ββ24αγ)=y1𝐵superscript2𝛼1𝛽superscript𝛽24𝛼𝛾subscript𝑦1B=(2\alpha)^{-1}\left(-\beta-\sqrt{\beta^{2}-4\alpha\gamma}\right)=y_{1}. Note that y=B𝑦𝐵y=B is a fixed point of the system (2.3). We obtain the following modified discrete dynamical system [Solís and Jódar(2004)]

xn+1=g(xn)xn2b(λ)xn=xn(xnb(λ))subscript𝑥𝑛1𝑔subscript𝑥𝑛superscriptsubscript𝑥𝑛2𝑏𝜆subscript𝑥𝑛subscript𝑥𝑛subscript𝑥𝑛𝑏𝜆x_{n+1}=g(x_{n})\equiv x_{n}^{2}-b(\lambda)x_{n}=x_{n}(x_{n}-b(\lambda)) (2.4)

where b(λ)=β(λ)2α(λ)B𝑏𝜆𝛽𝜆2𝛼𝜆𝐵b(\lambda)=-\beta(\lambda)-2\alpha(\lambda)B. The map xn+1=g(xn)subscript𝑥𝑛1𝑔subscript𝑥𝑛x_{n+1}=g(x_{n}) is unimodal. Without loss of generality, we will restrict to the case of positive and continuous b(λ)𝑏𝜆b(\lambda) for λ>0𝜆0\lambda>0.

Under the above transformation, the fixed points of the new system are

x0=0,x1=1+b(λ)formulae-sequencesubscript𝑥00subscript𝑥11𝑏𝜆x_{0}=0,\quad x_{1}=1+b(\lambda) (2.5)

so that the eigenvalue function for each of them is given by

ϕ(x0)=gx(x0)=b(λ),ϕ(x1)=gx(x1)=2+b(λ).formulae-sequenceitalic-ϕsubscript𝑥0𝑔𝑥subscript𝑥0𝑏𝜆italic-ϕsubscript𝑥1𝑔𝑥subscript𝑥12𝑏𝜆\phi(x_{0})=\frac{\partial g}{\partial x}(x_{0})=-b(\lambda),\quad\phi(x_{1})=\frac{\partial g}{\partial x}(x_{1})=2+b(\lambda).

The corresponding regions of type 1 are

x0subscript𝑥0\displaystyle x_{0} ::\displaystyle: {(λ,b(λ))|1<b(λ)<1}conditional-set𝜆𝑏𝜆1𝑏𝜆1\displaystyle\{(\lambda,b(\lambda))|-1<b(\lambda)<1\}
x1subscript𝑥1\displaystyle x_{1} ::\displaystyle: {(λ,b(λ))|3<b(λ)<1}.conditional-set𝜆𝑏𝜆3𝑏𝜆1\displaystyle\{(\lambda,b(\lambda))|-3<b(\lambda)<-1\}.

There are also two periodic points of period two, which we can calculate by solving

x=g(2)(x)=g(g(x))=x42bx3+b(b1)x2+b2x.𝑥superscript𝑔2𝑥𝑔𝑔𝑥superscript𝑥42𝑏superscript𝑥3𝑏𝑏1superscript𝑥2superscript𝑏2𝑥x=g^{(2)}(x)=g(g(x))=x^{4}-2bx^{3}+b(b-1)x^{2}+b^{2}x.

We can factor x(x(b+1))𝑥𝑥𝑏1x(x-(b+1)) from the expression since it corresponds to the fixed points (order 1) and we then solve for the roots of

x2+(1b)x+(1b)=0;superscript𝑥21𝑏𝑥1𝑏0x^{2}+(1-b)x+(1-b)=0;

and, again by the general quadratic formula, we have

x12superscriptsubscript𝑥12\displaystyle x_{1}^{2} =\displaystyle= (2)1(b1+b2+2b3)superscript21𝑏1superscript𝑏22𝑏3\displaystyle(2)^{-1}\left(b-1+\sqrt{b^{2}+2b-3}\right)
x22superscriptsubscript𝑥22\displaystyle x_{2}^{2} =\displaystyle= (2)1(b1b2+2b3)superscript21𝑏1superscript𝑏22𝑏3\displaystyle(2)^{-1}\left(b-1-\sqrt{b^{2}+2b-3}\right)

with eigenvalue function

ϕ2(λ)=g(2)x(x)=4x36bx2+2b(b1)x+b2|x=xj2,j=1,2formulae-sequencesubscriptitalic-ϕ2𝜆superscript𝑔2𝑥𝑥4superscript𝑥36𝑏superscript𝑥22𝑏𝑏1𝑥evaluated-atsuperscript𝑏2𝑥superscriptsubscript𝑥𝑗2𝑗12\phi_{2}(\lambda)=\frac{\partial g^{(2)}}{\partial x}(x)=\left.4x^{3}-6bx^{2}+2b(b-1)x+b^{2}\right|_{x=x_{j}^{2}},\quad j=1,2

and, therefore, corresponding type 2 region given by

{(λ,b(λ))|1<b(λ)<611.45}.conditional-set𝜆𝑏𝜆1𝑏𝜆611.45\{(\lambda,b(\lambda))|1<b(\lambda)<\sqrt{6}-1\approx 1.45\}.

Moreover, there are 2Nsuperscript2𝑁2^{N} periodic points of period 2Nsuperscript2𝑁2^{N} with region of type N𝑁N given by

{(λ,b(λ))|bN1<b(λ)<bN}conditional-set𝜆𝑏𝜆subscript𝑏𝑁1𝑏𝜆subscript𝑏𝑁\{(\lambda,b(\lambda))|b_{N-1}<b(\lambda)<b_{N}\}

where the sequence {bk}k=0superscriptsubscriptsubscript𝑏𝑘𝑘0\{b_{k}\}_{k=0}^{\infty} is convergent, and converges to the limiting value b=1.569945subscript𝑏1.569945b_{\infty}=1.569945\cdots [Solís and Jódar(2004)]. The sequence {bk}k=0superscriptsubscriptsubscript𝑏𝑘𝑘0\{b_{k}\}_{k=0}^{\infty} is related to the one described by [Feigenbaum(1978)] in [Feigenbaum(1978)] and converges exponentially. Table 2.2 summarizes this results.

Type of the region (k𝑘k) Interval [(bk,bk+1)subscript𝑏𝑘subscript𝑏𝑘1(b_{k},b_{k+1})]
1 (x0subscript𝑥0x_{0}) (-3, -1)
1 (x1subscript𝑥1x_{1}) (-1, 1)
2 (x12,x22superscriptsubscript𝑥12superscriptsubscript𝑥22x_{1}^{2},x_{2}^{2}) (1, 611.45)\sqrt{6}-1\approx 1.45)
\vdots \vdots
\infty (b=1.56994567,subscript𝑏1.56994567b_{\infty}=1.56994567...,\infty)
Table 2.2: Regions of type k𝑘k for the quadratic map xn+1=xn2b(λ)xnsubscript𝑥𝑛1superscriptsubscript𝑥𝑛2𝑏𝜆subscript𝑥𝑛x_{n+1}=x_{n}^{2}-b(\lambda)x_{n}.

The values of the original Feigenbaum sequence, {λk}k=0superscriptsubscriptsubscript𝜆𝑘𝑘0\{\lambda_{k}\}_{k=0}^{\infty}, correspond to λ𝜆\lambda values of the points of intersection of the function b(λ)𝑏𝜆b(\lambda) with the the values of the sequence {bk}k=0superscriptsubscriptsubscript𝑏𝑘𝑘0\{b_{k}\}_{k=0}^{\infty}. Stated otherwise, the sequence {bk}k=0superscriptsubscriptsubscript𝑏𝑘𝑘0\{b_{k}\}_{k=0}^{\infty} results from evaluating the Feigenbaum sequence {λk}k=0superscriptsubscriptsubscript𝜆𝑘𝑘0\{\lambda_{k}\}_{k=0}^{\infty} in the function b(λ)𝑏𝜆b(\lambda). When the function b(λ)𝑏𝜆b(\lambda) is the identity function, we can recover the original Feigenbaum’s sequence.

Feigenbaum’s sequence, {λk}k=0superscriptsubscriptsubscript𝜆𝑘𝑘0\{\lambda_{k}\}_{k=0}^{\infty}, describes the values of the parameter λ𝜆\lambda for which the period doubling bifurcation takes place in a unimodal map. The sequence converges exponentially. The values {λk}k=0superscriptsubscriptsubscript𝜆𝑘𝑘0\{\lambda_{k}\}_{k=0}^{\infty} for a given iteration function g𝑔g –of the following form– are given implicitly by the definition of [Feigenbaum(1978)] in its original paper [Feigenbaum(1978)], for a map f𝑓f as

xp2k=g2k(xp2k)=(λkf)2k(xp2k),λ,formulae-sequencesuperscriptsubscript𝑥𝑝superscript2𝑘superscript𝑔superscript2𝑘superscriptsubscript𝑥𝑝superscript2𝑘superscriptsubscript𝜆𝑘𝑓superscript2𝑘superscriptsubscript𝑥𝑝superscript2𝑘𝜆x_{p}^{2^{k}}=g^{2^{k}}(x_{p}^{2^{k}})=(\lambda_{k}f)^{2^{k}}(x_{p}^{2^{k}}),\quad\lambda\in\mathbb{N},

for the periodic point of period 2ksuperscript2𝑘2^{k}, xp2ksuperscriptsubscript𝑥𝑝superscript2𝑘x_{p}^{2^{k}}, where we must solve for λksubscript𝜆𝑘\lambda_{k}. In table 2.4 the values of a Feigenbaum sequence are shown for a specific iteration function.

A simple way to estimate the values of Feigenbaum sequences is starting from Feigenbaum’s constant, δ=4.669201609𝛿4.669201609\delta=4.669201609..., and the first calculated values of the sequence {λk}ksubscriptsubscript𝜆𝑘𝑘\{\lambda_{k}\}_{k\in\mathbb{N}}, through the approximation

λk+1=λk+λkλk1δ.subscript𝜆𝑘1subscript𝜆𝑘subscript𝜆𝑘subscript𝜆𝑘1𝛿\lambda_{k+1}=\lambda_{k}+\frac{\lambda_{k}-\lambda_{k-1}}{\delta}. (2.6)

This estimate is used as a predictor for the next value of the sequence and tends to be more precise as k𝑘k\rightarrow\infty.

Refer to caption
Figure 2.4: Feigenbaum’s sequence for f(x)=12x2𝑓𝑥12superscript𝑥2f(x)=1-2x^{2}. N𝑁N denotes the cycle of order 2Nsuperscript2𝑁2^{N}. The parameter is denoted as λ𝜆\lambda. δN=(λN+1λN)/(λN+2λN+1)subscript𝛿𝑁subscript𝜆𝑁1subscript𝜆𝑁subscript𝜆𝑁2subscript𝜆𝑁1\delta_{N}=(\lambda_{N+1}-\lambda_{N})/(\lambda_{N+2}-\lambda_{N+1}) and δN=(δN+1δN)/(δN+2δN+1)superscriptsubscript𝛿𝑁subscript𝛿𝑁1subscript𝛿𝑁subscript𝛿𝑁2subscript𝛿𝑁1\delta_{N}^{\prime}=(\delta_{N+1}-\delta_{N})/(\delta_{N+2}-\delta_{N+1}). Reproduced from [Feigenbaum(1978)]

2.3 Types of periodic points and chaos

By the known properties of the sequence {bk}k=0superscriptsubscriptsubscript𝑏𝑘𝑘0\{b_{k}\}_{k=0}^{\infty} and Sarkovskii’s theorem, we can state the following propositions [Solís and Jódar(2004)]:

Proposition 2.1.

If the function b(λ)𝑏𝜆b(\lambda) is bounded from above by bNsubscript𝑏𝑁b_{N}, the map xn+1=g(xn)xn2b(λ)xnsubscript𝑥𝑛1𝑔subscript𝑥𝑛superscriptsubscript𝑥𝑛2𝑏𝜆subscript𝑥𝑛x_{n+1}=g(x_{n})\equiv x_{n}^{2}-b(\lambda)x_{n} can only have periodic points of period 2msuperscript2𝑚2^{m} with m<N𝑚𝑁m<N. Moreover, if the bound is given by bsubscript𝑏b_{\infty}, the system is not chaotic and it can only have periodic points of periods of powers of two.

It is worth remarking that there are unimodal maps that have no periodic points with periods greater than one, in which case the bifurcation diagram consists of a single branch given by the fixed point.

Proposition 2.2.

Suppose that the function b(λ)𝑏𝜆b(\lambda) is bounded. If the supreme of the function b(λ)𝑏𝜆b(\lambda) lies within the interval (bn,bn+1)subscript𝑏𝑛subscript𝑏𝑛1(b_{n},\,b_{n+1}), then the map xn+1=xn2b(λ)xnsubscript𝑥𝑛1superscriptsubscript𝑥𝑛2𝑏𝜆subscript𝑥𝑛x_{n+1}=x_{n}^{2}-b(\lambda)x_{n} only has periodic points of period 2ksuperscript2𝑘2^{k} with k{1, 2,,n}𝑘12𝑛k\in\{1,\,2,\,...,\,n\} and therefore, it is not chaotic.

In order for the system xn+1=xn2b(λ)xnsubscript𝑥𝑛1superscriptsubscript𝑥𝑛2𝑏𝜆subscript𝑥𝑛x_{n+1}=x_{n}^{2}-b(\lambda)x_{n} to be a regular reversal map for λ𝜆\lambda in some interval 𝒜=(λI,λF)𝒜subscript𝜆𝐼subscript𝜆𝐹\mathcal{A}=(\lambda_{I},\,\lambda_{F}), it is necessary that the function b(λ)𝑏𝜆b(\lambda) has the following property: there exist λ1<λ2<λ3subscript𝜆1subscript𝜆2subscript𝜆3\lambda_{1}<\lambda_{2}<\lambda_{3} in 𝒜𝒜\mathcal{A} such that b(λ1)<b(λ3)=bi<b(λ2)𝑏subscript𝜆1𝑏subscript𝜆3subscript𝑏𝑖𝑏subscript𝜆2b(\lambda_{1})<b(\lambda_{3})=b_{i}<b(\lambda_{2}) for some i𝑖i\in\mathbb{N} where bisubscript𝑏𝑖b_{i} is the i𝑖i-th element of the sequence {bk}k=0superscriptsubscriptsubscript𝑏𝑘𝑘0\{b_{k}\}_{k=0}^{\infty}. It is relatively easy to construct functions b(λ)𝑏𝜆b(\lambda) with the aforementioned property, which means there is a large class of unimodal maps that are not chaotic.

Beyond the value bsubscript𝑏b_{\infty}, there are other regions that divide the positive quadrant of 2superscript2\mathbb{R}^{2}. The upper region of 2superscript2\mathbb{R}^{2}, that is, the region {(λ,b(λ))|b(λ)>b}conditional-set𝜆𝑏𝜆𝑏𝜆subscript𝑏\{(\lambda,b(\lambda))|b(\lambda)>b_{\infty}\}, can be defined as the region of type \infty and corresponds to divergent orbits. It is not clear how this region is subdivided and, in general, this is an open question.

2.4 Quadratic examples

The goal of the following examples is to show that regular reversal maps are a very broad family and that it is relatively simple to construct maps with specific –ad hoc– bifurcation diagrams with or without chaos.

Example 2.3.

Let b(λ)𝑏𝜆b(\lambda) be a quadratic unimodal function given by

b(λ)=4α0β02λ(β0λ),𝑏𝜆4subscript𝛼0superscriptsubscript𝛽02𝜆subscript𝛽0𝜆b(\lambda)=\frac{4\alpha_{0}}{\beta_{0}^{2}}\lambda(\beta_{0}-\lambda),

with α0,β0>0subscript𝛼0subscript𝛽00\alpha_{0},\,\beta_{0}>0. Note that α0subscript𝛼0\alpha_{0} is the maximum value and β0subscript𝛽0\beta_{0} is a root of b(λ)𝑏𝜆b(\lambda) (the other root is λ=0𝜆0\lambda=0). We are interested in knowing the periodic point structure of the map xn+1=xn2b(λ)xnsubscript𝑥𝑛1superscriptsubscript𝑥𝑛2𝑏𝜆subscript𝑥𝑛x_{n+1}=x_{n}^{2}-b(\lambda)x_{n} as we vary the parameter α0subscript𝛼0\alpha_{0}. Fixing β0=2subscript𝛽02\beta_{0}=2, we then have the following cases:

  • If 0<α0<b0=10subscript𝛼0subscript𝑏010<\alpha_{0}<b_{0}=1, we have a bifurcation diagram consisting of a single straight line coincident with the λ𝜆\lambda axis.

  • If 1<α0<b2=611subscript𝛼0subscript𝑏2611<\alpha_{0}<b_{2}=\sqrt{6}-1, we obtain a regular reversal map with a bifurcation diagram consisting of a straight line, as the previous case, but with a closed asymmetrical loop homeomorphic to a circle (see figure 2.6a on page 2.6a). The lack of symmetry stems from the fact that the upper branch of the loop corresponds the the periodic point x12superscriptsubscript𝑥12x_{1}^{2} and the lower branch to x22superscriptsubscript𝑥22x_{2}^{2}. The diameter of the loop is calculated by finding the two intersections of b(λ)𝑏𝜆b(\lambda) with the line b=b1=1𝑏subscript𝑏11b=b_{1}=1 and it is equal to 2β0α021α02subscript𝛽0superscriptsubscript𝛼021subscript𝛼02\beta_{0}\sqrt{\alpha_{0}^{2}-1}\alpha_{0}. The graph of b(λ)𝑏𝜆b(\lambda) is shown in figure 2.5a at page 2.5a.

  • If b1=1<α0<61=b2subscript𝑏11subscript𝛼061subscript𝑏2b_{1}=1<\alpha_{0}<\sqrt{6}-1=b_{2}, two new loops appear: one that comes out of the lower branch of the previous case and another one from the upper branch, corresponding to the periodic points of period 22superscript222^{2} (see figure 2.6b). The matching graph of b(λ)𝑏𝜆b(\lambda) is shown in figure 2.5b.

  • If bn<α0<bn+1subscript𝑏𝑛subscript𝛼0subscript𝑏𝑛1b_{n}<\alpha_{0}<b_{n+1} for some n𝑛n\in\mathbb{N}, the graph of b(λ)𝑏𝜆b(\lambda) looks like in figure 2.5c and we have a bifurcation diagram consisting of a straight line with a set of nested loops as shown in figure 2.6c.

  • If α0=1.56994567=bsubscript𝛼01.56994567subscript𝑏\alpha_{0}=1.56994567...=b_{\infty}, then we have branches that bifurcate to form infinite nested loops, as is shown in figure 2.6d and whose graph of b(λ)𝑏𝜆b(\lambda) is shown in figure 2.5d.

Refer to caption
(a) α0=1.2subscript𝛼01.2\alpha_{0}=1.2 and β0=2subscript𝛽02\beta_{0}=2.
Refer to caption
(b) α0=1.5subscript𝛼01.5\alpha_{0}=1.5 and β0=2subscript𝛽02\beta_{0}=2.
Refer to caption
(c) α0=1.562subscript𝛼01.562\alpha_{0}=1.562 and β0=2subscript𝛽02\beta_{0}=2.
Refer to caption
(d) α0=b1.57subscript𝛼0subscript𝑏1.57\alpha_{0}=b_{\infty}\approx 1.57 and β0=2subscript𝛽02\beta_{0}=2.
Figure 2.5: b(λ)𝑏𝜆b(\lambda) functions of the quadratic regular reversal maps of example 2.3. The horizontal lines correspond to the values of bksubscript𝑏𝑘b_{k}, for k=1,2𝑘12k=1,2 and \infty.
Refer to caption
(a) α0=1.2subscript𝛼01.2\alpha_{0}=1.2 y β0=2subscript𝛽02\beta_{0}=2.
Refer to caption
(b) b2<α0=1.5<b3subscript𝑏2subscript𝛼01.5subscript𝑏3b_{2}<\alpha_{0}=1.5<b_{3} and β0=2subscript𝛽02\beta_{0}=2.
Refer to caption
(c) b3<α0=1.562<b4subscript𝑏3subscript𝛼01.562subscript𝑏4b_{3}<\alpha_{0}=1.562<b_{4} y β0=2subscript𝛽02\beta_{0}=2.
Refer to caption
(d) α0=1.57bsubscript𝛼01.57subscript𝑏\alpha_{0}=1.57\approx b_{\infty} and β0=2subscript𝛽02\beta_{0}=2.
Figure 2.6: Bifurcation diagrams of the quadratic regular reversal maps of example 2.3.
Example 2.4.

If we modify b(λ)𝑏𝜆b(\lambda) choosing it as periodic “unimodal”222Formally, this cannot be a unimodal function, since it is a requirement for a unimodal function to have a single maximum of the form

b(λ)=αsin2π(λ+3/2)+1,𝑏𝜆𝛼2𝜋𝜆321b(\lambda)=\alpha\sin{2\pi(\lambda+3/2)}+1, (2.7)

with 1+α<b1𝛼subscript𝑏1+\alpha<b_{\infty}), the new bifurcation diagrams will be very similar to those of the last case, except that they will be periodic. In figure 2.7 at page 2.7, the graphs of b(λ)𝑏𝜆b(\lambda) are shown, and in figure 2.8, are the corresponding bifurcation diagrams, both for the corresponding parameter cases of figures 2.5 and 2.6, respectively.

Refer to caption
(a) α=0.2𝛼0.2\alpha=0.2.
Refer to caption
(b) α=0.5𝛼0.5\alpha=0.5.
Refer to caption
(c) α=0.562𝛼0.562\alpha=0.562.
Refer to caption
(d) α=0.57𝛼0.57\alpha=0.57. (1+αb1𝛼subscript𝑏1+\alpha\approx b_{\infty}.)
Figure 2.7: Functions b(λ)𝑏𝜆b(\lambda) of the regular reversal maps of example 2.4 given by the equation 2.4. The horizontal lines correspond to the values of bksubscript𝑏𝑘b_{k}, for k=1,2𝑘12k=1,2 and \infty.
Refer to caption
(a) α=0.2𝛼0.2\alpha=0.2.
Refer to caption
(b) b2<1+α=1.5<b3subscript𝑏21𝛼1.5subscript𝑏3b_{2}<1+\alpha=1.5<b_{3}.
Refer to caption
(c) b3<1+α=1.562<b4subscript𝑏31𝛼1.562subscript𝑏4b_{3}<1+\alpha=1.562<b_{4}.
Refer to caption
(d) 1+α=1.57b1𝛼1.57subscript𝑏1+\alpha=1.57\approx b_{\infty}.
Figure 2.8: Bifurcation the regular reversal maps of example 2.4.
Example 2.5.

If we introduce an asymmetry factor to the function 2.7 we can achieve an asymmetrical bifurcation diagram with the different lobes having different amplitudes. One way to accomplish the above is defining b(λ)𝑏𝜆b(\lambda) as

b(λ)=(1eλ)sin2(πλ/4){αsin[π(λ+3/2)]+4/5}+4/5.𝑏𝜆1superscript𝑒𝜆superscript2𝜋𝜆4𝛼𝜋𝜆324545b(\lambda)=\left(1-e^{-\lambda}\right)\sin^{2}\left(\pi\lambda/4\right)\{\alpha\sin\left[\pi\left(\lambda+3/2\right)\right]+4/5\}+4/5. (2.8)

Varying the parameter α𝛼\alpha we obtain graphs of b(λ)𝑏𝜆b(\lambda) like the ones shown in figure 2.9 at page 2.9, with its respective bifurcation diagrams, shown in figure 2.10 at page 2.10. For small values of α𝛼\alpha we have two lobes (or “humps”) in the function b(λ)𝑏𝜆b(\lambda), with its local maxima under the value b2subscript𝑏2b_{2}, which translates into a bifurcation diagram with four asymmetrical lobes and of different amplitude (see figure 2.10a at page 2.10a). However, due to asymmetry, rising the value of α𝛼\alpha to 0.40.40.4 will only produce the appearance of nested loops in two of the right lobes of the bifurcation diagram (see figure 2.10b), which repeats itself as we continue to raise the value of α𝛼\alpha (see figure 2.10c).

Refer to caption
(a) α=0.2𝛼0.2\alpha=0.2.
Refer to caption
(b) α=0.4𝛼0.4\alpha=0.4.
Refer to caption
(c) α=0.5𝛼0.5\alpha=0.5.
Figure 2.9: Graphs of b(λ)𝑏𝜆b(\lambda), given by equation 2.8, for the regular reverse maps of example 2.8.
Refer to caption
(a) α=0.2𝛼0.2\alpha=0.2.
Refer to caption
(b) α=0.4𝛼0.4\alpha=0.4.
Refer to caption
(c) α=0.5𝛼0.5\alpha=0.5.
Figure 2.10: Bifurcation diagrams of the regular reversal maps of example 2.5.
Example 2.6.

A similar effect to the one of the last example can be achieved by taking b(λ)𝑏𝜆b(\lambda) as a polynomial of fourth grade in λ𝜆\lambda, with its coefficients chosen in such way that it is not unimodal. For example,

b(λ)=(αλ41.533λ3+4.1083λ24.166λ)𝑏𝜆𝛼superscript𝜆41.533superscript𝜆34.1083superscript𝜆24.166𝜆b(\lambda)=-\left(\alpha\lambda^{4}-1.533\lambda^{3}+4.1083\lambda^{2}-4.166\lambda\right) (2.9)

with α(0.18,0.192)𝛼0.180.192\alpha\in(0.18,0.192). The function b(λ)𝑏𝜆b(\lambda) has three local extremes (see figure 2.11 at page 2.11). In this case, the bifurcation diagram is shown for different values of the parameter in figure 2.12 at page 2.12 where, for progressively higher values of α𝛼\alpha we progressively obtain lobes that stem from the branches corresponding to the periodic points. In this case the diagram is no symmetrical nor periodic because the function b(λ)𝑏𝜆b(\lambda) is neither.

Refer to caption
(a) α=0.19166𝛼0.19166\alpha=0.19166.
Refer to caption
(b) α=0.191𝛼0.191\alpha=0.191.
Refer to caption
(c) α=0.19𝛼0.19\alpha=0.19.
Figure 2.11: Graphs of b(λ)𝑏𝜆b(\lambda), given by equation 2.9, for the quadratic regular reversal maps of example 2.6.
Refer to caption
(a) α=0.19166𝛼0.19166\alpha=0.19166.
Refer to caption
(b) α=0.191𝛼0.191\alpha=0.191.
Refer to caption
(c) α=0.19𝛼0.19\alpha=0.19.
Figure 2.12: Bifurcation diagrams of the regular reversal maps of example 2.6.

With the above examples we have shown we can design discrete dynamical systems in order to have specific bifurcation diagrams “on demand” with desired properties. The fundamental fact that allows us to do the latter is to note where the function ϕ(λ)italic-ϕ𝜆\phi(\lambda) intersects the values of the sequence {bk}k=1superscriptsubscriptsubscript𝑏𝑘𝑘1\{b_{k}\}_{k=1}^{\infty}. The detailed form of ϕ(λ)italic-ϕ𝜆\phi(\lambda) is irrelevant as long as it does not cross a value of the sequence, for it only affects the specific form of the “branches” of the periodic points in the bifurcation diagram, but it does not alter the fundamental structure of the set of periodic points.

Chapter 3 Quadratic maps revisited

Now, we will restate and expand the work of [Solís and Jódar(2004)] in a way that will allow us to gain greater insight into the way we can control the appearance of chaos in a parametric family of quadratic maps and we will be able to generalize these results in a natural way for cubic maps and, later on, for general polynomial maps of n𝑛n-th degree.

3.1 General quadratic form

Consider the quadratic iteration function f(y)𝑓𝑦f(y) as stated by the

Definition 3.1 (General Quadratic Map).

The General Quadratic Map (GQM) is defined by

f2(y):=α+(β+1)y+γy2=yPf2(y),assignsubscript𝑓2𝑦𝛼𝛽1𝑦𝛾superscript𝑦2𝑦subscript𝑃subscript𝑓2𝑦f_{2}(y):=\alpha+(\beta+1)y+\gamma y^{2}=y-P_{f_{2}}(y), (3.1)

where

Pf2(y):=(α+βy+γy2)assignsubscript𝑃subscript𝑓2𝑦𝛼𝛽𝑦𝛾superscript𝑦2P_{f_{2}}(y):=-(\alpha+\beta y+\gamma y^{2}) (3.2)

and α𝛼\alpha, β𝛽\beta and γ𝛾\gamma are real coefficients that depend upon the parameter λ𝜆\lambda and are C1superscript𝐶1C^{1}. Pf2subscript𝑃subscript𝑓2P_{f_{2}} is called the Fixed Points Polynomial (of the general quadratic map).

It is clear that any quadratic map can be put in this form. Although the minus sign in the definition of Pf2subscript𝑃subscript𝑓2P_{f_{2}} may now seem “unnatural”, its function will be clear shortly111Check page 3.2 below..

Evidently, the zeros of Pf2subscript𝑃subscript𝑓2P_{f_{2}} are the fixed points of f2subscript𝑓2f_{2} –thence the name of Pf2subscript𝑃subscript𝑓2P_{f_{2}}– since, if y0subscript𝑦0y_{0} is a root of Pf2subscript𝑃subscript𝑓2P_{f_{2}}, then Pf2(y0)=0subscript𝑃subscript𝑓2subscript𝑦00P_{f_{2}}(y_{0})=0 implies f2(y0)=y0subscript𝑓2subscript𝑦0subscript𝑦0f_{2}(y_{0})=y_{0}.

In the following, for simplicity, we will drop the subscript “2” from f𝑓f to denote the CQM. Since the derivatives of f𝑓f are

f(y)superscript𝑓𝑦\displaystyle f^{\prime}(y) =2γy+β+1,absent2𝛾𝑦𝛽1\displaystyle=2\gamma y+\beta+1, (3.3)
f′′(y)superscript𝑓′′𝑦\displaystyle f^{\prime\prime}(y) =2γ,absent2𝛾\displaystyle=2\gamma,

straightforward calculations lead us to

Lemma 3.1.

f𝑓f has the following two roots and critical point, respectively,

y0rsuperscriptsubscript𝑦0𝑟\displaystyle y_{0}^{r} =β+1+(β+1)24αγ2γ,absent𝛽1superscript𝛽124𝛼𝛾2𝛾\displaystyle=-\frac{\beta+1+\sqrt{(\beta+1)^{2}-4\alpha\gamma}}{2\gamma}, (3.4)
y1rsuperscriptsubscript𝑦1𝑟\displaystyle y_{1}^{r} =β+1(β+1)24αγ2γ,absent𝛽1superscript𝛽124𝛼𝛾2𝛾\displaystyle=-\frac{\beta+1-\sqrt{(\beta+1)^{2}-4\alpha\gamma}}{2\gamma},
ycsubscript𝑦𝑐\displaystyle y_{c} =β+12γ,absent𝛽12𝛾\displaystyle=-\frac{\beta+1}{2\gamma},

with corresponding extreme value

fext=4αγ(β+1)24γ=αγyc.subscript𝑓𝑒𝑥𝑡4𝛼𝛾superscript𝛽124𝛾𝛼𝛾subscript𝑦𝑐f_{ext}=\frac{4\alpha\gamma-(\beta+1)^{2}}{4\gamma}=\alpha-\gamma y_{c}.

And, finally, it has the fixed points

y0superscriptsubscript𝑦0\displaystyle y_{0}^{*} =β+β24αγ2γ,absent𝛽superscript𝛽24𝛼𝛾2𝛾\displaystyle=-\frac{\beta+\sqrt{{\beta}^{2}-4\,\alpha\,\gamma}}{2\,\gamma}, (3.5)
y1superscriptsubscript𝑦1\displaystyle y_{1}^{*} =ββ24αγβ2γ.absent𝛽superscript𝛽24𝛼𝛾𝛽2𝛾\displaystyle=-\frac{\beta-\sqrt{{\beta}^{2}-4\,\alpha\,\gamma}-\beta}{2\,\gamma}.

3.2 Linear factors form

Suppose we know both roots of Pf2subscript𝑃subscript𝑓2P_{f_{2}} and they are y0,y1subscript𝑦0subscript𝑦1y_{0},\,y_{1}\in\mathbb{C}, which exist by the fundamental theorem of algebra. Then, by the factor theorem, we know that (yyi),i{0,1}𝑦subscript𝑦𝑖𝑖01(y-y_{i}),\,i\in\{0,1\} is a factor of Pf2subscript𝑃subscript𝑓2P_{f_{2}} and we can rewrite it as

Pf2(y)=M(yy0)(yy1),M{0}.formulae-sequencesubscript𝑃subscript𝑓2𝑦𝑀𝑦subscript𝑦0𝑦subscript𝑦1𝑀0P_{f_{2}}(y)=M(y-y_{0})(y-y_{1}),\quad M\in\mathbb{R}\setminus\{0\}. (3.6)

Therefore, we can also rewrite f2subscript𝑓2f_{2} as

f2(y)=yM(yy0)(yy1).subscript𝑓2𝑦𝑦𝑀𝑦subscript𝑦0𝑦subscript𝑦1f_{2}(y)=y-M(y-y_{0})(y-y_{1}). (3.7)

Since M{0}𝑀0M\in\mathbb{R}\setminus\{0\}, then we can define M:=sM~assign𝑀𝑠~𝑀M:=s\tilde{M}, where

s:=sgn(M)={1,ifM>0,1,ifM<0,assign𝑠sgn𝑀cases1if𝑀01if𝑀0s:=\mathrm{sgn}(M)=\begin{cases}1,&\mathrm{if}\,M>0,\\ -1,&\mathrm{if}\,M<0,\end{cases} (3.8)

is the sign function and M~:=|M|assign~𝑀𝑀\tilde{M}:=|M|; this definition will be useful for our purposes further below.

With the above, we have

Definition 3.2 (Linear Factors Form of the Quadratic Map).

The GQM can be rewritten as

f2(y)=ysM~(yy0)(yy1):=h2(y,λ)subscript𝑓2𝑦𝑦𝑠~𝑀𝑦subscript𝑦0𝑦subscript𝑦1assignsubscript2𝑦𝜆f_{2}(y)=y-s\tilde{M}(y-y_{0})(y-y_{1}):=h_{2}(y,\,\lambda) (3.9)

where M,y1𝑀subscript𝑦1M,\,y_{1} and y2subscript𝑦2y_{2} are smooth functions of the parameter λ𝜆\lambda. We call h2subscript2h_{2} the Linear Factors Form of the Quadratic Map (LFFQM).

Again, straightforward calculations lead us to

Lemma 3.2.

(3.9) has first and second derivatives with respect to y𝑦y given by

h2(y,λ)superscriptsubscript2𝑦𝜆\displaystyle h_{2}^{\prime}(y,\,\lambda) =1sM~(2yy0y1),absent1𝑠~𝑀2𝑦subscript𝑦0subscript𝑦1\displaystyle=1-s\tilde{M}\left(2y-y_{0}-y_{1}\right), (3.10)
h2′′(y,λ)superscriptsubscript2′′𝑦𝜆\displaystyle h_{2}^{\prime\prime}(y,\,\lambda) =2sM~.absent2𝑠~𝑀\displaystyle=-2\,s\tilde{M}.

The roots of h2subscript2h_{2} are

y0rsuperscriptsubscript𝑦0𝑟\displaystyle y_{0}^{r} =s2M~2(y1y0)2+2sM~(y1+y0)+1sM~(y1+y0)12sM~absentsuperscript𝑠2superscript~𝑀2superscriptsubscript𝑦1subscript𝑦022𝑠~𝑀subscript𝑦1subscript𝑦01𝑠~𝑀subscript𝑦1subscript𝑦012𝑠~𝑀\displaystyle=-\frac{\sqrt{s^{2}\tilde{M}^{2}(y_{1}-y_{0})^{2}+2s\tilde{M}(y_{1}+y_{0})+1}-s\tilde{M}(y_{1}+y_{0})-1}{2s\tilde{M}} (3.11)
y1rsuperscriptsubscript𝑦1𝑟\displaystyle y_{1}^{r} =s2M~2(y1y0)2+2sM~(y1+y0)+1+sM~(y1+y0)+12sM~absentsuperscript𝑠2superscript~𝑀2superscriptsubscript𝑦1subscript𝑦022𝑠~𝑀subscript𝑦1subscript𝑦01𝑠~𝑀subscript𝑦1subscript𝑦012𝑠~𝑀\displaystyle=\frac{\sqrt{s^{2}\tilde{M}^{2}(y_{1}-y_{0})^{2}+2s\tilde{M}(y_{1}+y_{0})+1}+s\tilde{M}(y_{1}+y_{0})+1}{2s\tilde{M}}

and the only critical point is at

yc=sM~(y1+y0)+12sM~,subscript𝑦𝑐𝑠~𝑀subscript𝑦1subscript𝑦012𝑠~𝑀y_{c}=\frac{s\tilde{M}(y_{1}+y_{0})+1}{2s\tilde{M}},

with corresponding extreme value

hext=y0+[sM~(y1y0)+1]24sM~.subscript𝑒𝑥𝑡subscript𝑦0superscriptdelimited-[]𝑠~𝑀subscript𝑦1subscript𝑦0124𝑠~𝑀h_{ext}=y_{0}+\frac{[s\tilde{M}(y_{1}-y_{0})+1]^{2}}{4s\tilde{M}}.

Here the apparently “unnatural” minus sign of the definition 3.1 of the fixed points polynomial Pf2subscript𝑃subscript𝑓2P_{f_{2}} will show its usefulness: for purely aesthetic reasons, in the definition of the LFFQM, we want that h2(y0)>0superscriptsubscript2subscript𝑦00h_{2}^{\prime}(y_{0})>0 whenever 0<y0<y10subscript𝑦0subscript𝑦10<y_{0}<y_{1} and M>0𝑀0M>0 which, without loss of generality, is accomplished by arbitrarily introducing the mentioned sign.

3.3 Canonical form

We will introduce a canonical form of the quadratic map by requiring both fixed points in the linear factors form (3.2) to be real or, equivalently, β24αγ>0superscript𝛽24𝛼𝛾0\beta^{2}-4\alpha\gamma>0 in the general form (3.1); also, we will require an “amplitude”, M(λ)𝑀𝜆M(\lambda), equal to unity222Recall M(λ)𝑀𝜆M(\lambda) is the function preceding the product of the linear factors of the fixed points in the LFFQM defined in Eq. (3.2)., a fixed point mapped to zero333This is the reason we require both fixed points to be real, otherwise we would need a complex transformation to map the complex fixed point to zero, which is beyond the scope of this work. and the other fixed point being mapped to

x1(λ):=sM~(y1y0).assignsubscript𝑥1𝜆𝑠~𝑀subscript𝑦1subscript𝑦0x_{1}(\lambda):=s\tilde{M}(y_{1}-y_{0}). (3.12)

The latter is accomplished by taking one of the following linear transformations

y=T0(x)𝑦subscript𝑇0𝑥\displaystyle y=T_{0}(x) :=y0+sM~1x,assignabsentsubscript𝑦0𝑠superscript~𝑀1𝑥\displaystyle:=y_{0}+s\tilde{M}^{-1}x, (3.13)
y=T1(x)𝑦subscript𝑇1𝑥\displaystyle y=T_{1}(x) :=y1+sM~1x.assignabsentsubscript𝑦1𝑠superscript~𝑀1𝑥\displaystyle:=y_{1}+s\tilde{M}^{-1}x.

Without loss of generality, we will take transformation T0subscript𝑇0T_{0} and call it simply T𝑇T. Since T𝑇T is evidently linear, its inverse can be calculated directly as

x=T1(y)=sM~(yy0).𝑥superscript𝑇1𝑦𝑠~𝑀𝑦subscript𝑦0x=T^{-1}(y)=s\tilde{M}(y-y_{0}). (3.14)

Performing the corresponding calculations on the LFFQM we obtain

Definition 3.3 (Canonical Quadratic Map).

The map defined by

g2(x,λ):=xx(xx1(λ)),assignsubscript𝑔2𝑥𝜆𝑥𝑥𝑥subscript𝑥1𝜆g_{2}(x,\lambda):=x-x(x-x_{1}(\lambda)), (3.15)

where λ𝜆\lambda\in\mathbb{R} is a parameter, is called the Canonical Quadratic Map (CQM).

We have thus mapped the real fixed points y00maps-tosubscript𝑦00y_{0}\mapsto 0 and y1x1maps-tosubscript𝑦1subscript𝑥1y_{1}\mapsto x_{1}. This form is much simpler than the previous ones and it has the advantage of putting the whole parametric dependence onto the single non-constant fixed point x1=x1(λ)subscript𝑥1subscript𝑥1𝜆x_{1}=x_{1}(\lambda) through (3.12). Implicit in this dependence is the parametric dependence of the corresponding original roots of the LFFQM (3.9), so we can explicitly give this dependence if desired. Moreover, the fact that T𝑇T is actually a topological conjugacy between h2subscript2h_{2} and g2subscript𝑔2g_{2} will be proved in chapter 5, by which we can state that the study of stability and chaos in quadratic maps with real fixed points can be summarized by the study of the Canonical Quadratic Map, which yields its importance.

With some straightforward calculations, we easily arrive at the following

Lemma 3.3.

The Canonical Quadratic Map has the roots

x0rsuperscriptsubscript𝑥0𝑟\displaystyle x_{0}^{r} =0,absent0\displaystyle=0, (3.16)
x1rsuperscriptsubscript𝑥1𝑟\displaystyle x_{1}^{r} =x1+1,absentsubscript𝑥11\displaystyle=x_{1}+1,

and has derivatives

g2(x,λ)superscriptsubscript𝑔2𝑥𝜆\displaystyle g_{2}^{\prime}(x,\lambda) =2x+x1+1,absent2𝑥subscript𝑥11\displaystyle=-2x+x_{1}+1, (3.17)
g2′′(x,λ)superscriptsubscript𝑔2′′𝑥𝜆\displaystyle g_{2}^{\prime\prime}(x,\lambda) =2,absent2\displaystyle=-2,

so that its critical point is at

xc=x1+12subscript𝑥𝑐subscript𝑥112x_{c}=\frac{x_{1}+1}{2}

with value

xm=(x1+1)24=xc2.subscript𝑥𝑚superscriptsubscript𝑥1124superscriptsubscript𝑥𝑐2x_{m}=\frac{(x_{1}+1)^{2}}{4}=x_{c}^{2}. (3.18)

We see then that the second root of the canonical map, x1rsuperscriptsubscript𝑥1𝑟x_{1}^{r}, is always one unit to the right of the non-constant fixed point x1subscript𝑥1x_{1}. Also, the critical point is always a maximum and is located in the middle point between 1 and x1subscript𝑥1x_{1}, with its value being simply the square of the point.

3.4 Stability and chaos in the canonical quadratic map

3.4.1 Fixed points

The stability of the fixed points is given by the absolute value of its multiplier, which must be less than one. Making use of the defined eigenvalue function, we have from (3.17) that for the fixed point x0=0subscript𝑥00x_{0}=0,

ϕ0(λ)=g2(0)=x1(λ)+1subscriptitalic-ϕ0𝜆superscriptsubscript𝑔20subscript𝑥1𝜆1\phi_{0}(\lambda)=g_{2}^{\prime}(0)=x_{1}(\lambda)+1 (3.19)

so that it is straightforward to see that if

2<x1(λ)<02subscript𝑥1𝜆0-2<x_{1}(\lambda)<0 (SC0)

then x0=0subscript𝑥00x_{0}=0 is a stable fixed point. We will call inequality SC0 the stability condition for the fixed point x0=0subscript𝑥00x_{0}=0. We see then that the stability of x0subscript𝑥0x_{0} actually depends on the value of x1subscript𝑥1x_{1}. In terms of the original fixed points of the quadratic map, y0subscript𝑦0y_{0} and y1subscript𝑦1y_{1}, what counts is the separation between them scaled by the factor sM~𝑠~𝑀s\tilde{M}, as can be seen from equation (3.12).

On the other hand, for x1subscript𝑥1x_{1},

ϕ1(λ)=g2(x1)=1x1subscriptitalic-ϕ1𝜆superscriptsubscript𝑔2subscript𝑥11subscript𝑥1\phi_{1}(\lambda)=g_{2}^{\prime}(x_{1})=1-x_{1} (3.20)

so that x1subscript𝑥1x_{1} is stable as long as

0<x1(λ)<2.0subscript𝑥1𝜆20<x_{1}(\lambda)<2. (SC1)

We call the value b1=2subscript𝑏12b_{1}=2 our first bifurcation value. Notice that b1=2subscript𝑏12-b_{1}=-2 is also a bifurcation value since it is the limit of the stability condition for x0=0subscript𝑥00x_{0}=0, given by (SC0). We summarize this results in the following

Proposition 3.4.

Let x0=0subscript𝑥00x_{0}=0 and x1subscript𝑥1x_{1} be the two fixed points of the canonical quadratic map as defined above. Then

  • if 2<x1<02subscript𝑥10-2<x_{1}<0, zero will be an asymptotically stable fixed point;

  • if 0<x1<20subscript𝑥120<x_{1}<2, x1subscript𝑥1x_{1} will be an asymptotically stable fixed point.

3.4.2 Periodic points of period two

To determine the period two periodic points we must of course solve

g22(x)=x.superscriptsubscript𝑔22𝑥𝑥g_{2}^{2}(x)=x.

Some basic algebra shows that

g22(x)=(x1+1)2x(x1+1)(x1+2)x2+2(x1+1)x3x4.superscriptsubscript𝑔22𝑥superscriptsubscript𝑥112𝑥subscript𝑥11subscript𝑥12superscript𝑥22subscript𝑥11superscript𝑥3superscript𝑥4g_{2}^{2}(x)=(x_{1}+1)^{2}x-(x_{1}+1)(x_{1}+2)x^{2}+2(x_{1}+1)x^{3}-x^{4}.

Therefore, we must solve

x1(x1+2)x(x1+1)(x1+2)x2+2(x1+1)x3x4=0.subscript𝑥1subscript𝑥12𝑥subscript𝑥11subscript𝑥12superscript𝑥22subscript𝑥11superscript𝑥3superscript𝑥40x_{1}(x_{1}+2)x-(x_{1}+1)(x_{1}+2)x^{2}+2(x_{1}+1)x^{3}-x^{4}=0.

Clearly, we must factor out x=0𝑥0x=0 and x=x1𝑥subscript𝑥1x=x_{1} since we know those are the fixed points of the original map, and we are looking (only) for period two points. Performing the factorization, we get

x2(x1+2)x+(x1+2)=0.superscript𝑥2subscript𝑥12𝑥subscript𝑥120x^{2}-(x_{1}+2)x+(x_{1}+2)=0. (3.21)

Solving the latter equation we get to the following

Lemma 3.5.

The canonical quadratic map has a unique 2-cycle given by

x02=12[x1+2+(x1)24],x12=12[x1+2(x1)24],formulae-sequencesuperscriptsubscript𝑥0212delimited-[]subscript𝑥12superscriptsubscript𝑥124superscriptsubscript𝑥1212delimited-[]subscript𝑥12superscriptsubscript𝑥124x_{0}^{2}=\frac{1}{2}\left[x_{1}+2+\sqrt{(x_{1})^{2}-4}\right],\quad x_{1}^{2}=\frac{1}{2}\left[x_{1}+2-\sqrt{(x_{1})^{2}-4}\right],

where the superscript will indicate the period of the periodic point.

Proof.

It is straightforward to see that g2(x02)=x12subscript𝑔2superscriptsubscript𝑥02superscriptsubscript𝑥12g_{2}(x_{0}^{2})=x_{1}^{2} and g2(x12)=x02subscript𝑔2superscriptsubscript𝑥12superscriptsubscript𝑥02g_{2}(x_{1}^{2})=x_{0}^{2}, therefore proving that both x02superscriptsubscript𝑥02x_{0}^{2} and x12superscriptsubscript𝑥12x_{1}^{2} are period 2 periodic points of g2subscript𝑔2g_{2} and the orbit {x02,x12}superscriptsubscript𝑥02superscriptsubscript𝑥12\{x_{0}^{2},\,x_{1}^{2}\} is indeed a 2-cycle. The uniqueness comes from the fact that both are algebraic roots of the second degree equation (3.21) and every such period 2 periodic point must be so. ∎

To analyze the stability of this 2-cycle, we must calculate its own eigenvalue function. The derivative of g22superscriptsubscript𝑔22g_{2}^{2} is

g22x(x)=(x1+1)22(x1+1)(x1+2)x+6(x1+1)x24x3,superscriptsubscript𝑔22𝑥𝑥superscriptsubscript𝑥1122subscript𝑥11subscript𝑥12𝑥6subscript𝑥11superscript𝑥24superscript𝑥3\frac{\partial g_{2}^{2}}{\partial x}(x)=(x_{1}+1)^{2}-2(x_{1}+1)(x_{1}+2)x+6(x_{1}+1)x^{2}-4x^{3},

so that evaluating in x=x02(λ)𝑥superscriptsubscript𝑥02𝜆x=x_{0}^{2}(\lambda) and x=x12(λ)𝑥superscriptsubscript𝑥12𝜆x=x_{1}^{2}(\lambda) we have

(g22)(x02(λ))=(g22)(x12(λ))=ϕ2(λ)=5x1(λ)2,superscriptsubscriptsuperscript𝑔22superscriptsubscript𝑥02𝜆superscriptsubscriptsuperscript𝑔22superscriptsubscript𝑥12𝜆subscriptitalic-ϕ2𝜆5subscript𝑥1superscript𝜆2\left(g^{2}_{2}\right)^{\prime}(x_{0}^{2}(\lambda))=\left(g^{2}_{2}\right)^{\prime}(x_{1}^{2}(\lambda))=\phi_{2}(\lambda)=5-x_{1}(\lambda)^{2}, (3.22)

so that, not surprisingly, both points of the 2-cycle have the same stability criteria. The stability region is then determined by

|5(x1)2|<15superscriptsubscript𝑥121|5-(x_{1})^{2}|<1

whose solutions lead us to

Lemma 3.6.

The unique 2-cycle of the canonical quadratic map is asymptotically stable whenever one of the following inequalities is met

66\displaystyle-\sqrt{6} <x1absentsubscript𝑥1\displaystyle<x_{1} <2,absent2\displaystyle<-2,
22\displaystyle 2 <x1absentsubscript𝑥1\displaystyle<x_{1} <6,absent6\displaystyle<\sqrt{6},

where x1subscript𝑥1x_{1} is the nonzero fixed point of the CQM.

So, we see that the stability region is divided in two in terms of the value of the original fixed point x1subscript𝑥1x_{1}. As a reference, 62.4494962.44949\sqrt{6}\approx 2.44949. We call this value b2=6subscript𝑏26b_{2}=\sqrt{6} the second bifurcation value of the parameter λ𝜆\lambda for the canonical quadratic map. Notice again that b2=6subscript𝑏26-b_{2}=-\sqrt{6} is also a bifurcation value.

It is worth noting that in the case of the first inequality of lemma 3.6, it is the zero fixed point that produces a bifurcation, while for the second one, it is x1subscript𝑥1x_{1} that bifurcates to produce the 2-cycle. The latter can be seen from lemma 3.5 since, for |x1|<2subscript𝑥12|x_{1}|<2, the 2-cycle is not real and it only begins to be so at the bifurcation values ±b1plus-or-minussubscript𝑏1\pm b_{1}; when x1=2subscript𝑥12x_{1}=-2, both x02superscriptsubscript𝑥02x_{0}^{2} and x12superscriptsubscript𝑥12x_{1}^{2} are zero (and equal to the zero fixed point) and thereon separate as the two branches of the square root function (positive and negative). On the other hand, when x1=2subscript𝑥12x_{1}=2, both x02superscriptsubscript𝑥02x_{0}^{2} and x12superscriptsubscript𝑥12x_{1}^{2} are also 2 in value and thereon separate in the same way as before.

We can now prove that, indeed, what happens at the bifurcation values of ±b1=±2plus-or-minussubscript𝑏1plus-or-minus2\pm b_{1}=\pm 2 are period doubling bifurcations.

Proposition 3.7.

Let444The dot represents derivative with respect to the parameter λ𝜆\lambda to avoid confusion with the apostrophe representing derivative with respect to x𝑥x. x1(λ0)=2subscript𝑥1subscript𝜆02x_{1}(\lambda_{0})=-2 (respectively, x1(λ0)=2subscript𝑥1subscript𝜆02x_{1}(\lambda_{0})=2) and x1˙(λ0)0˙subscript𝑥1subscript𝜆00\dot{x_{1}}(\lambda_{0})\neq 0. Then the zero fixed point (respectively, the x1subscript𝑥1x_{1} fixed point) of the CQM undergoes a period doubling bifurcation precisely when λ=λ0𝜆subscript𝜆0\lambda=\lambda_{0}.

Proof.

According to theorem 1.12 we must prove that:

  1. 1.

    g2(0)=0subscript𝑔200g_{2}(0)=0 for all λ𝜆\lambda in an interval around λ=2𝜆2\lambda=-2. This is trivially true since the zero fixed point does not vary with the parameter.

  2. 2.

    g2(0)=1superscriptsubscript𝑔201g_{2}^{\prime}(0)=-1. Which follows directly from eq. 3.19.

  3. 3.

    (g22)λ|λ=2(0)0evaluated-atsuperscriptsubscriptsuperscript𝑔22𝜆𝜆200\left.\frac{\partial\left(g^{2}_{2}\right)^{\prime}}{\partial\lambda}\right|_{\lambda=-2}(0)\neq 0. From (3.22) we see that this is true since x1˙(λ)0˙subscript𝑥1𝜆0\dot{x_{1}}(\lambda)\neq 0 by hypothesis.

The case for the period doubling bifurcation of the other fixed point when x1(λ)=2subscript𝑥1𝜆2x_{1}(\lambda)=2 is analogous. ∎

The requirement x1˙(λ)0˙subscript𝑥1𝜆0\dot{x_{1}}(\lambda)\neq 0 will be understood more clearly in the examples below. Likewise, we can prove that, in turn, the fixed point x02superscriptsubscript𝑥02x_{0}^{2} (or x12superscriptsubscript𝑥12x_{1}^{2}) of g22superscriptsubscript𝑔22g_{2}^{2} —periodic point of period 2 for g2subscript𝑔2g_{2}— also undergoes a period doubling bifurcation associated with the second bifurcation value when x1(λ)=±b2=±6subscript𝑥1𝜆plus-or-minussubscript𝑏2plus-or-minus6x_{1}(\lambda)=\pm b_{2}=\pm\sqrt{6}.

Proposition 3.8.

Let x1(λ0)=6subscript𝑥1subscript𝜆06x_{1}(\lambda_{0})=-\sqrt{6} and x1˙(λ0)0˙subscript𝑥1subscript𝜆00\dot{x_{1}}(\lambda_{0})\neq 0. Then the fixed points x02superscriptsubscript𝑥02x_{0}^{2} and x12superscriptsubscript𝑥12x_{1}^{2} of the g22superscriptsubscript𝑔22g_{2}^{2} undergo a period doubling bifurcation precisely when λ=λ0𝜆subscript𝜆0\lambda=\lambda_{0}.

Proof.

Consider first the case for x02superscriptsubscript𝑥02x_{0}^{2} given by lemma 3.5. According to theorem 1.12 we must prove that:

  1. 1.

    (g22)(x02)=1superscriptsuperscriptsubscript𝑔22superscriptsubscript𝑥021\left(g_{2}^{2}\right)^{\prime}(x_{0}^{2})=-1. Which follows directly from eq. 3.22 since by hypothesis x1(λ0)=6subscript𝑥1subscript𝜆06x_{1}(\lambda_{0})=-\sqrt{6}.

  2. 2.

    g22(x02)=x02superscriptsubscript𝑔22superscriptsubscript𝑥02superscriptsubscript𝑥02g_{2}^{2}(x_{0}^{2})=x_{0}^{2} for all λ𝜆\lambda in an interval around λ=6𝜆6\lambda=-\sqrt{6}. Theorem 1.11 allows us to affirm this since by the above point (g22)(x02)1superscriptsuperscriptsubscript𝑔22superscriptsubscript𝑥021\left(g_{2}^{2}\right)^{\prime}(x_{0}^{2})\neq 1.

  3. 3.

    (g24)λ|λ=λ0(x02)0evaluated-atsuperscriptsubscriptsuperscript𝑔42𝜆𝜆subscript𝜆0superscriptsubscript𝑥020\left.\frac{\partial\left(g^{4}_{2}\right)^{\prime}}{\partial\lambda}\right|_{\lambda=\lambda_{0}}(x_{0}^{2})\neq 0. Since, by the chain rule,

    (g24(x02))superscriptsuperscriptsubscript𝑔24superscriptsubscript𝑥02\displaystyle\left(g_{2}^{4}(x_{0}^{2})\right)^{\prime} =(g22(g22(x02)))absentsuperscriptsuperscriptsubscript𝑔22superscriptsubscript𝑔22superscriptsubscript𝑥02\displaystyle=\left(g_{2}^{2}\left(g_{2}^{2}(x_{0}^{2})\right)\right)^{\prime}
    =(g22)(g22(x02))(g22(x02))absentsuperscriptsuperscriptsubscript𝑔22superscriptsubscript𝑔22superscriptsubscript𝑥02superscriptsuperscriptsubscript𝑔22superscriptsubscript𝑥02\displaystyle=\left(g_{2}^{2}\right)^{\prime}\left(g_{2}^{2}(x_{0}^{2})\right)\,\left(g_{2}^{2}(x_{0}^{2})\right)^{\prime}
    =(g22(x02))(1(x1)2)absentsuperscriptsuperscriptsubscript𝑔22superscriptsubscript𝑥021superscriptsubscript𝑥12\displaystyle=\left(g_{2}^{2}(x_{0}^{2})\right)^{\prime}\left(1-(x_{1})^{2}\right)
    =(1(x1)2)2absentsuperscript1superscriptsubscript𝑥122\displaystyle=\left(1-(x_{1})^{2}\right)^{2}

    using eq. (3.22) and the fact that x02superscriptsubscript𝑥02x_{0}^{2} is a fixed point of g22superscriptsubscript𝑔22g_{2}^{2}. The partial derivative with respect to λ𝜆\lambda is then

    (g24)λ(x02(λ0))superscriptsubscriptsuperscript𝑔42𝜆superscriptsubscript𝑥02subscript𝜆0\displaystyle\frac{\partial\left(g^{4}_{2}\right)^{\prime}}{\partial\lambda}(x_{0}^{2}(\lambda_{0})) =4(1(x1(λ0))2)(x1(λ0))2x1˙(λ0)absent41superscriptsubscript𝑥1subscript𝜆02superscriptsubscript𝑥1subscript𝜆02˙subscript𝑥1subscript𝜆0\displaystyle=-4\left(1-(x_{1}(\lambda_{0}))^{2}\right)(x_{1}(\lambda_{0}))^{2}\dot{x_{1}}(\lambda_{0})
    =120x1˙(λ0)0absent120˙subscript𝑥1subscript𝜆00\displaystyle=120\,\dot{x_{1}}(\lambda_{0})\neq 0

    since x1(λ0)=6subscript𝑥1subscript𝜆06x_{1}(\lambda_{0})=-\sqrt{6} and x1˙(λ0)0˙subscript𝑥1subscript𝜆00\dot{x_{1}}(\lambda_{0})\neq 0 by hypothesis.

The cases for x12superscriptsubscript𝑥12x_{1}^{2} and λ0=6subscript𝜆06\lambda_{0}=-\sqrt{6} are analogous. ∎

3.4.3 Periodic points of higher period

Some more algebra shows us that

g23(x)=x(x13+3x12+3x1+1)+x2(x145x1310x129x13)+x3(2x14+10x13+20x12+18x1+6)+x4(x1410x1325x1225x19)+x5(4x13+18x12+24x1+10)+x6(6x1214x18)+x7(4x1+4)x8superscriptsubscript𝑔23𝑥𝑥superscriptsubscript𝑥133superscriptsubscript𝑥123subscript𝑥11superscript𝑥2superscriptsubscript𝑥145superscriptsubscript𝑥1310superscriptsubscript𝑥129subscript𝑥13superscript𝑥32superscriptsubscript𝑥1410superscriptsubscript𝑥1320superscriptsubscript𝑥1218subscript𝑥16superscript𝑥4superscriptsubscript𝑥1410superscriptsubscript𝑥1325superscriptsubscript𝑥1225subscript𝑥19superscript𝑥54superscriptsubscript𝑥1318superscriptsubscript𝑥1224subscript𝑥110superscript𝑥66superscriptsubscript𝑥1214subscript𝑥18superscript𝑥74subscript𝑥14superscript𝑥8g_{2}^{3}(x)=x\,\left({x_{1}}^{3}+3\,{x_{1}}^{2}+3\,x_{1}+1\right)+{x}^{2}\,\left(-{x_{1}}^{4}-5\,{x_{1}}^{3}-10\,{x_{1}}^{2}-9\,x_{1}-3\right)\\ +{x}^{3}\,\left(2\,{x_{1}}^{4}+10\,{x_{1}}^{3}+20\,{x_{1}}^{2}+18\,x_{1}+6\right)+{x}^{4}\,\left(-{x_{1}}^{4}-10\,{x_{1}}^{3}-25\,{x_{1}}^{2}-25\,x_{1}-9\right)\\ +{x}^{5}\,\left(4\,{x_{1}}^{3}+18\,{x_{1}}^{2}+24\,x_{1}+10\right)+{x}^{6}\,\left(-6\,{x_{1}}^{2}-14\,x_{1}-8\right)+{x}^{7}\,\left(4\,x_{1}+4\right)-{x}^{8} (3.23)

with derivative

(g23(x))=(x13+3x12+3x1+1)+x(2x1410x1320x1218x16)+x2(6x14+30x13+60x12+54x1+18)+x3(4x1440x13100x12100x136)+x4(20x13+90x12+120x1+50)+x5(36x1284x148)+x6(28x1+28)8x7superscriptsuperscriptsubscript𝑔23𝑥superscriptsubscript𝑥133superscriptsubscript𝑥123subscript𝑥11𝑥2superscriptsubscript𝑥1410superscriptsubscript𝑥1320superscriptsubscript𝑥1218subscript𝑥16superscript𝑥26superscriptsubscript𝑥1430superscriptsubscript𝑥1360superscriptsubscript𝑥1254subscript𝑥118superscript𝑥34superscriptsubscript𝑥1440superscriptsubscript𝑥13100superscriptsubscript𝑥12100subscript𝑥136superscript𝑥420superscriptsubscript𝑥1390superscriptsubscript𝑥12120subscript𝑥150superscript𝑥536superscriptsubscript𝑥1284subscript𝑥148superscript𝑥628subscript𝑥1288superscript𝑥7\left(g_{2}^{3}(x)\right)^{\prime}=\left({x_{1}}^{3}+3\,{x_{1}}^{2}+3\,x_{1}+1\right)+x\,\left(-2\,{x_{1}}^{4}-10\,{x_{1}}^{3}-20\,{x_{1}}^{2}-18\,x_{1}-6\right)\\ +{x}^{2}\,\left(6\,{x_{1}}^{4}+30\,{x_{1}}^{3}+60\,{x_{1}}^{2}+54\,x_{1}+18\right)+{x}^{3}\,\left(-4\,{x_{1}}^{4}-40\,{x_{1}}^{3}-100\,{x_{1}}^{2}-100\,x_{1}-36\right)\\ +{x}^{4}\left(20\,{x_{1}}^{3}+90\,{x_{1}}^{2}+120\,x_{1}+50\right)+{x}^{5}\,\left(-36\,{x_{1}}^{2}-84\,x_{1}-48\right)+{x}^{6}\,\left(28\,x_{1}+28\right)-8\,{x}^{7} (3.24)

So that, in order to find the periodic points of period three we must, in principle, explicitly solve

x7+4(x1+1)x6(6x12+14x1+8)x5+(4x13+18x12+24x1+10)x4(x14+10x13+25x12+25x1+9)x3+(2x14+10x13+20x12+18x1+6)x2(x14+5x13+10x12+9x1+3)x+(x13+3x12+3x1+1)=0superscript𝑥74subscript𝑥11superscript𝑥66superscriptsubscript𝑥1214subscript𝑥18superscript𝑥54superscriptsubscript𝑥1318superscriptsubscript𝑥1224subscript𝑥110superscript𝑥4superscriptsubscript𝑥1410superscriptsubscript𝑥1325superscriptsubscript𝑥1225subscript𝑥19superscript𝑥32superscriptsubscript𝑥1410superscriptsubscript𝑥1320superscriptsubscript𝑥1218subscript𝑥16superscript𝑥2superscriptsubscript𝑥145superscriptsubscript𝑥1310superscriptsubscript𝑥129subscript𝑥13𝑥superscriptsubscript𝑥133superscriptsubscript𝑥123subscript𝑥110-x^{7}+4(x_{1}+1)\,x^{6}-(6\,x_{1}^{2}+14\,x_{1}+8)\,x^{5}+(4\,x_{1}^{3}+18\,x_{1}^{2}+24\,x_{1}+10)\,x^{4}\\ -(x_{1}^{4}+10\,x_{1}^{3}+25\,x_{1}^{2}+25\,x_{1}+9)\,x^{3}+(2\,x_{1}^{4}+10\,x_{1}^{3}+20\,x_{1}^{2}+18\,x_{1}+6)\,x^{2}\\ -(x_{1}^{4}+5\,x_{1}^{3}+10\,x_{1}^{2}+9\,x_{1}+3)\,x+(x_{1}^{3}+3\,x_{1}^{2}+3\,x_{1}+1)=0 (3.25)

which is an seventh degree polynomial in x𝑥x. Even factoring out the other known fixed point of g2subscript𝑔2g_{2}, x1subscript𝑥1x_{1}, we are still left with a sixth degree polynomial on x𝑥x which is, in general, impossible to solve explicitly by algebraic means. Nevertheless, for specific polynomials we can determine the roots numerically.

Much less can be said about determining higher period periodic points. However, we may still approximate the stability regions of these k𝑘k-cycles by numerically determining the bifurcation value of the parameter λ𝜆\lambda, which in this case we have done by plotting very high-precision bifurcation diagrams (see Appendix A). One way to determine an approximation to the value of the infinite period bifurcation value, after which the onset of chaos takes place, is through equation (2.6), which yields a very rough estimate of b2.57subscript𝑏2.57b_{\infty}\approx 2.57. Some bifurcation values for the CQM found in this work are shown in table 3.1; moreover, table 3.2 shows the the stability conditions for each periodic point,up to period 27superscript272^{7}; the boundary of the stability bands correspond to the bksubscript𝑏𝑘b_{k} bifurcation values.

k𝑘k bksubscript𝑏𝑘b_{k}
0 0
1 2
2 66\sqrt{6}
3 2.5440±0.0005plus-or-minus2.54400.00052.5440\pm 0.0005
4 2.5642±0.0002plus-or-minus2.56420.00022.5642\pm 0.0002
5 2.56871±4×105plus-or-minus2.568714superscript1052.56871\pm 4\times 10^{-5}
6 2.56966±1×105plus-or-minus2.569661superscript1052.56966\pm 1\times 10^{-5}
7 2.569881±5×106plus-or-minus2.5698815superscript1062.569881\pm 5\times 10^{-6}
\vdots \vdots
\infty 2.569941±5×107similar-toabsentplus-or-minus2.5699415superscript107\sim 2.569941\pm 5\times 10^{-7}
Table 3.1: Bifurcation values for the Canonical Quadratic Map. b0=0subscript𝑏00b_{0}=0 is included only as a reference, although it is not a bifurcation value.
Period Periodic Points Stability Condition
1 x01=0superscriptsubscript𝑥010x_{0}^{1}=0 b1<x1<b0subscript𝑏1subscript𝑥1subscript𝑏0-b_{1}<x_{1}<b_{0}
1 x11=x1superscriptsubscript𝑥11subscript𝑥1x_{1}^{1}=x_{1} b0<x1<b1subscript𝑏0subscript𝑥1subscript𝑏1b_{0}<x_{1}<b_{1}
2 x02=12[x1+2+(x1)24]superscriptsubscript𝑥0212delimited-[]subscript𝑥12superscriptsubscript𝑥124x_{0}^{2}=\frac{1}{2}\left[x_{1}+2+\sqrt{(x_{1})^{2}-4}\right] b1<|x1|<b2subscript𝑏1subscript𝑥1subscript𝑏2b_{1}<|x_{1}|<b_{2}
x12=12[x1+2(x1)24]superscriptsubscript𝑥1212delimited-[]subscript𝑥12superscriptsubscript𝑥124x_{1}^{2}=\frac{1}{2}\left[x_{1}+2-\sqrt{(x_{1})^{2}-4}\right]
4 x04,x14,x24,x34superscriptsubscript𝑥04superscriptsubscript𝑥14superscriptsubscript𝑥24superscriptsubscript𝑥34x_{0}^{4},\,x_{1}^{4},\,x_{2}^{4},\,x_{3}^{4} b2<|x1|<b3subscript𝑏2subscript𝑥1subscript𝑏3b_{2}<|x_{1}|<b_{3}
8 x08,x18,,x78superscriptsubscript𝑥08superscriptsubscript𝑥18superscriptsubscript𝑥78x_{0}^{8},\,x_{1}^{8},\,...,x_{7}^{8} b3<|x1|<b4subscript𝑏3subscript𝑥1subscript𝑏4b_{3}<|x_{1}|<b_{4}
16 x016,x116,,x1516superscriptsubscript𝑥016superscriptsubscript𝑥116superscriptsubscript𝑥1516x_{0}^{16},\,x_{1}^{16},\,...,x_{15}^{16} b4<|x1|<b5subscript𝑏4subscript𝑥1subscript𝑏5b_{4}<|x_{1}|<b_{5}
32 x032,x132,,x3132superscriptsubscript𝑥032superscriptsubscript𝑥132superscriptsubscript𝑥3132x_{0}^{32},\,x_{1}^{32},\,...,x_{31}^{32} b5<|x1|<b6subscript𝑏5subscript𝑥1subscript𝑏6b_{5}<|x_{1}|<b_{6}
64 x064,x164,,x6364superscriptsubscript𝑥064superscriptsubscript𝑥164superscriptsubscript𝑥6364x_{0}^{64},\,x_{1}^{64},\,...,x_{63}^{64} b6<|x1|<b7subscript𝑏6subscript𝑥1subscript𝑏7b_{6}<|x_{1}|<b_{7}
\infty - |x1|>bsubscript𝑥1subscript𝑏|x_{1}|>b_{\infty}
Table 3.2: Periodic points and corresponding stability conditions for the canonical quadratic map.

3.4.4 Fixed points with multiplicity

In the CQM, the only way we can have multiplicity in the fixed points is when x1(λ)=x0=0subscript𝑥1𝜆subscript𝑥00x_{1}(\lambda)=x_{0}=0. In this case, g2subscript𝑔2g_{2} takes the form

g2(x)=x(1x).subscript𝑔2𝑥𝑥1𝑥g_{2}(x)=x\,(1-x). (3.26)

Its derivative is then

g2(x)=12x,superscriptsubscript𝑔2𝑥12𝑥g_{2}^{\prime}(x)=1-2x, (3.27)

and, therefore, g2(0)=1superscriptsubscript𝑔201g_{2}^{\prime}(0)=1, so that it is a nonhyperbolic fixed point, according to definition 1.6.

Proposition 3.9.

Let g2subscript𝑔2g_{2} be the CQM with a single fixed point with multiplicity of two, as in eq. (3.26). Then the single fixed point x0=x1=0subscript𝑥0subscript𝑥10x_{0}=x_{1}=0 is semistable from the right.

Proof.

It is straightforward to see that g2′′(x)=2superscriptsubscript𝑔2′′𝑥2g_{2}^{\prime\prime}(x)=-2 and then g2′′′(x)=0superscriptsubscript𝑔2′′′𝑥0g_{2}^{\prime\prime\prime}(x)=0 for all x𝑥x. Therefore, in particular, g2′′(0)=20superscriptsubscript𝑔2′′020g_{2}^{\prime\prime}(0)=-2\neq 0 and then g2′′′(0)=0superscriptsubscript𝑔2′′′00g_{2}^{\prime\prime\prime}(0)=0, so that, by the latter, we cannot use theorem 1.7. However, theorem 1.9 tells us that the fixed point is semistable from the right. Indeed, as g2subscript𝑔2g_{2} is concave downward at x=0𝑥0x=0, then g2superscriptsubscript𝑔2g_{2}^{\prime} is decreasing in a small neighborhood about zero and, since g2(0)=1superscriptsubscript𝑔201g_{2}^{\prime}(0)=1 and g2superscriptsubscript𝑔2g_{2}^{\prime} is continuous, we have that there exists δ>0𝛿0\delta>0, so that g2(x)>1superscriptsubscript𝑔2𝑥1g_{2}^{\prime}(x)>1 for x(δ, 0)𝑥𝛿 0x\in(-\delta,\,0) and g2(x)<1superscriptsubscript𝑔2𝑥1g_{2}^{\prime}(x)<1 for x(0,δ)𝑥0𝛿x\in(0,\,\delta). Consequently, x0=x1=0subscript𝑥0subscript𝑥10x_{0}=x_{1}=0 is unstable from the left and semistable from the right, according to definition 1.9 and the proof of theorem 1.1 [Elaydi(2000)]. ∎

It is worth noting that, although the case of multiplicity in the fixed point of the CQM is possible, the case of complex fixed points is not, since the CQM requires one fixed point to be zero, then its complex conjugate —which should also be a fixed point— is itself.

3.4.5 Regular-reversality and chaos in the canonical quadratic map

Let us recall, that the definition of x1subscript𝑥1x_{1} transforms, through T𝑇T, the parametric dependence of the original fixed points of the general quadratic map into a single parameter-dependent function x1=x1(λ)subscript𝑥1subscript𝑥1𝜆x_{1}=x_{1}(\lambda). Recall then that,

x1(λ)=M(λ)(y1(λ)y0(λ)).subscript𝑥1𝜆𝑀𝜆subscript𝑦1𝜆subscript𝑦0𝜆x_{1}(\lambda)=M(\lambda)\,\left(y_{1}(\lambda)-y_{0}(\lambda)\right).

Therefore, assuming a constant value of M(λ)=M0=1𝑀𝜆subscript𝑀01M(\lambda)=M_{0}=1, we see that the value of x1subscript𝑥1x_{1} is exactly the difference between the two original fixed points, y1subscript𝑦1y_{1} and y0subscript𝑦0y_{0}, of the General Quadratic Map (which we forced to be real). We can then interpret the found values of the sequence {bk}ksubscriptsubscript𝑏𝑘𝑘\{b_{k}\}_{k\in\mathbb{N}} as determining the

Definition 3.4 (Stability Bands of a Quadratic Map).

Let yi:𝒜:subscript𝑦𝑖𝒜y_{i}:\mathcal{A}\subseteq\mathbb{R}\rightarrow\mathbb{R}, i{1, 2}𝑖12i\in\{1,\,2\} be the two fixed points of the family of quadratic maps fλsubscript𝑓𝜆f_{\lambda}, as given by 3.1, and {bk}ksubscriptsubscript𝑏𝑘𝑘\{b_{k}\}_{k\in\mathbb{N}} the sequence of bifurcation values of table 3.1. The union of open intervals

(yi(λ)bk+1,yi(λ)bk)(yi(λ)+bk,yi(λ)+bk+1),λ𝒜subscript𝑦𝑖𝜆subscript𝑏𝑘1subscript𝑦𝑖𝜆subscript𝑏𝑘subscript𝑦𝑖𝜆subscript𝑏𝑘subscript𝑦𝑖𝜆subscript𝑏𝑘1𝜆𝒜\left(y_{i}(\lambda)-b_{k+1},\,y_{i}(\lambda)-b_{k}\right)\bigcup\left(y_{i}(\lambda)+b_{k},\,y_{i}(\lambda)+b_{k+1}\right),\quad\lambda\in\mathcal{A} (3.28)

is called the k𝑘k-th stability band of yisubscript𝑦𝑖y_{i}.

Graphically, we see then that stability bands run along the values of y1(λ)subscript𝑦1𝜆y_{1}(\lambda) and y0(λ)subscript𝑦0𝜆y_{0}(\lambda) as functions of the parameter and, as long as the value of one fixed point does not cross a the limit of a band of the other fixed point, there will be no changes in the periodic points structure of the map, i.e. there will only be bifurcations when the value of one fixed point crosses the limit of a stability band of the other fixed point. If M𝑀M were not held constant, it would simply act as a “modulator” for the width of the bands along the parameter λ𝜆\lambda.

Stability bands for x0=0subscript𝑥00x_{0}=0 and x1(λ)subscript𝑥1𝜆x_{1}(\lambda) in the CQM are defined analogously.

Definition 3.5 (Stability Bands of the CQM).

Let x1:𝒜:subscript𝑥1𝒜x_{1}:\mathcal{A}\subseteq\mathbb{R}\rightarrow\mathbb{R} be the nonzero fixed point of the family of quadratic maps g2subscript𝑔2g_{2}, as given by definition 3.3, and {bk}ksubscriptsubscript𝑏𝑘𝑘\{b_{k}\}_{k\in\mathbb{N}} the sequence of bifurcation values of table 3.1. The union of open intervals

(bk+1,bk)(bk,bk+1),λ𝒜subscript𝑏𝑘1subscript𝑏𝑘subscript𝑏𝑘subscript𝑏𝑘1𝜆𝒜\left(-b_{k+1},\,-b_{k}\right)\bigcup\left(b_{k},\,b_{k+1}\right),\quad\lambda\in\mathcal{A} (3.29)

is called the k𝑘k-th stability band of the CQM.

Notice that that the part of the k𝑘k-th stability band in the upper (respectively, lower) semi-plane corresponds to the stability condition of period 2ksuperscript2𝑘2^{k} attracting periodic points associated with period doubling bifurcation of the x1subscript𝑥1x_{1} (respectively, zero) fixed point.

Finally, we must remark that the width of these stability bands —given by the values of the sequence {bk}ksubscriptsubscript𝑏𝑘𝑘\{b_{k}\}_{k\in\mathbb{N}}— is not equal for polynomial maps of different degrees, as we shall see in the next chapters.

By the above, the big picture is that depending on the absolute value of x1(λ)subscript𝑥1𝜆x_{1}(\lambda) the one-parameter family of canonical quadratic maps can present or not attracting periodic points of different periods or even become chaotic. In particular, as long as |x1(λ)|<bNsubscript𝑥1𝜆subscript𝑏𝑁|x_{1}(\lambda)|<b_{N} we will only have attracting periodic points of period less or equal to 2Nsuperscript2𝑁2^{N}. We can thus restate the propositions 2.1 and 2.2 given in section 2.3 of chapter 2, originally by [Solís and Jódar(2004)] ([Solís and Jódar(2004)]).

Proposition 3.10.

If the absolute value of the fixed point x1(λ)superscriptsubscript𝑥1𝜆x_{1}^{*}(\lambda) is bounded from above by bNsubscript𝑏𝑁b_{N}, the GQM map xn+1=g2(xn)xnxn(xnx1(λ))subscript𝑥𝑛1subscript𝑔2subscript𝑥𝑛subscript𝑥𝑛subscript𝑥𝑛subscript𝑥𝑛superscriptsubscript𝑥1𝜆x_{n+1}=g_{2}(x_{n})\equiv x_{n}-x_{n}(x_{n}-x_{1}^{*}(\lambda)) is not chaotic and can only have periodic points of period 2msuperscript2𝑚2^{m} with m<N𝑚𝑁m<N. Moreover, if the bound is given by bsubscript𝑏b_{\infty}, the system is still not chaotic and it can only have periodic points of periods of powers of two.

And more specifically,

Proposition 3.11.

Suppose that the absolute value of the fixed point x1(λ)superscriptsubscript𝑥1𝜆x_{1}^{*}(\lambda) is bounded. If the supreme of the function |x1(λ)|superscriptsubscript𝑥1𝜆|x_{1}^{*}(\lambda)| lies within the interval (bn,bn+1)subscript𝑏𝑛subscript𝑏𝑛1(b_{n},\,b_{n+1}), then the map xn+1=g2(xn)xnxn(xnx1(λ))subscript𝑥𝑛1subscript𝑔2subscript𝑥𝑛subscript𝑥𝑛subscript𝑥𝑛subscript𝑥𝑛superscriptsubscript𝑥1𝜆x_{n+1}=g_{2}(x_{n})\equiv x_{n}-x_{n}(x_{n}-x_{1}^{*}(\lambda)) only has periodic points of period 2ksuperscript2𝑘2^{k} with k{1, 2,,n}𝑘12𝑛k\in\{1,\,2,\,...,\,n\} and therefore, it is not chaotic.

In order for the system xn+1=xnxn(xnx1(λ))subscript𝑥𝑛1subscript𝑥𝑛subscript𝑥𝑛subscript𝑥𝑛subscript𝑥1𝜆x_{n+1}=x_{n}-x_{n}\,\left(x_{n}-x_{1}(\lambda)\right) to be a regular reversal map for λ𝜆\lambda in some interval 𝒜=(λI,λF)𝒜subscript𝜆𝐼subscript𝜆𝐹\mathcal{A}=(\lambda_{I},\,\lambda_{F}), it is necessary that the continuous function x1(λ)subscript𝑥1𝜆x_{1}(\lambda) has the following property: there exist λ1<λ2<λ3subscript𝜆1subscript𝜆2subscript𝜆3\lambda_{1}<\lambda_{2}<\lambda_{3} in 𝒜𝒜\mathcal{A} such that x1(λ1)<x1(λ3)=bi<x1(λ2)subscript𝑥1subscript𝜆1subscript𝑥1subscript𝜆3subscript𝑏𝑖subscript𝑥1subscript𝜆2x_{1}(\lambda_{1})<x_{1}(\lambda_{3})=b_{i}<x_{1}(\lambda_{2}) for some i𝑖i\in\mathbb{N} where bisubscript𝑏𝑖b_{i} is the i𝑖i-th element of the sequence of bifurcation values {bk}k=0superscriptsubscriptsubscript𝑏𝑘𝑘0\{b_{k}\}_{k=0}^{\infty}. It is easy to construct functions x1(λ)subscript𝑥1𝜆x_{1}(\lambda) with the aforementioned property, which means that constructing regular-reversal canonical quadratic maps is also easy. Furthermore, knowing beforehand the properties of a desired bifurcation diagram, i.e. the fixed points, the λ𝜆\lambda values for desired bifurcations, etc. we can construct x1(λ)subscript𝑥1𝜆x_{1}(\lambda) functions that accomplish the desired properties in the bifurcation diagram.

3.5 Quadratic examples with the canonical map

Recall that the CQM is

g2(x,λ)=xx(xx1(λ)),subscript𝑔2𝑥𝜆𝑥𝑥𝑥subscript𝑥1𝜆g_{2}(x,\lambda)=x-x\left(x-x_{1}(\lambda)\right),

where

x1(λ)=M(λ)(y1(λ)y0(λ)).subscript𝑥1𝜆𝑀𝜆subscript𝑦1𝜆subscript𝑦0𝜆x_{1}(\lambda)=M(\lambda)\left(y_{1}(\lambda)-y_{0}(\lambda)\right). (3.30)

Therefore, we can “parametrize” the map g2(x,λ)subscript𝑔2𝑥𝜆g_{2}(x,\lambda) by defining either x1(λ)subscript𝑥1𝜆x_{1}(\lambda) directly or going back to the parametric dependence of the original LFFQM h2(x,λ)subscript2𝑥𝜆h_{2}(x,\lambda) from which x1subscript𝑥1x_{1} was defined using the transformation T𝑇T, i.e. defining M(λ)𝑀𝜆M(\lambda), y0(λ)subscript𝑦0𝜆y_{0}(\lambda) and y1(λ)subscript𝑦1𝜆y_{1}(\lambda). We will look at both interpretations at the same time in the following examples.

3.5.1 Linear fixed points

We will begin with the simplest case: linearly varying y0(λ)subscript𝑦0𝜆y_{0}(\lambda) and y1(λ)subscript𝑦1𝜆y_{1}(\lambda) and M(λ)=1𝑀𝜆1M(\lambda)=1. Consider,

y0(λ):=λ,y1(λ):=2λ.formulae-sequenceassignsubscript𝑦0𝜆𝜆assignsubscript𝑦1𝜆2𝜆y_{0}(\lambda):=\lambda,\quad y_{1}(\lambda):=2\,\lambda.

The graphs of these parametrizations of the fixed points are shown in figure 3.1. Clearly, x1(λ)=λsubscript𝑥1𝜆𝜆x_{1}(\lambda)=\lambda and its graph is also shown in figure 3.1.

Refer to caption
(a) Graphs of linear y0(λ)subscript𝑦0𝜆y_{0}(\lambda) and y1(λ)subscript𝑦1𝜆y_{1}(\lambda) along the selected values of λ𝜆\lambda. Also, the stability conditions (constrained by the bifurcation values b1subscript𝑏1b_{1}, b2subscript𝑏2b_{2} and bsubscript𝑏b_{\infty}) of y0subscript𝑦0y_{0} are shown around it.
Refer to caption
(b) Graph of corresponding x1(λ)subscript𝑥1𝜆x_{1}(\lambda) along with the bifurcation values b1subscript𝑏1b_{1}, b2subscript𝑏2b_{2} and bsubscript𝑏b_{\infty}.
Figure 3.1: Linear parametrization of fixed points (example 3.5.1).

The bifurcation diagram for this specific parametrization of the fixed points of the CQM is shown in figure 3.2. In this bifurcation diagram we can see that it is precisely at the values of λ𝜆\lambda for which y1subscript𝑦1y_{1} crosses limits of the stability conditions of y0subscript𝑦0y_{0} (or, equivalently, where x1subscript𝑥1x_{1} crosses the bifurcation values) that bifurcations take place. Notice that, since x1>0subscript𝑥10x_{1}>0, it is x1subscript𝑥1x_{1} the fixed point that undergoes period doubling bifurcation.

Refer to caption
Figure 3.2: Bifurcation diagram for the linear parametrization of the fixed points example.

In fact, it does not matter what the actual values of the slopes of y0(λ)subscript𝑦0𝜆y_{0}(\lambda) and y1(λ)subscript𝑦1𝜆y_{1}(\lambda) are: as long as their difference is the same (has the same slope) they will produce exactly the same bifurcation diagram, since the parametric dependence of x1(λ)subscript𝑥1𝜆x_{1}(\lambda) will be unchanged (see eq. 3.30). Making precise calculations of the bifurcations diagram near λ𝜆\lambda-values of interest we can determine numerically the rest (or at least some more) of the bifurcation values forming the sequence {bk}ksubscriptsubscript𝑏𝑘𝑘\{b_{k}\}_{k\in\mathbb{N}}, but this is best accomplished using the parametrization of example 3.5.5 below.

3.5.2 Quadratic fixed point

Redefining the parametric dependencies of y0subscript𝑦0y_{0} and y1subscript𝑦1y_{1} we can achieve regular-reversal maps as described in definition 2.6 in page 2.6. For example, leaving y0subscript𝑦0y_{0} in the above example 3.5.1, but redefining y1subscript𝑦1y_{1} with a quadratic parametric dependence on λ𝜆\lambda as

y0(λ)subscript𝑦0𝜆\displaystyle y_{0}(\lambda) :=λassignabsent𝜆\displaystyle:=\lambda (3.31)
y1(λ)subscript𝑦1𝜆\displaystyle y_{1}(\lambda) :=λp1λ(λp2),p1,p2.formulae-sequenceassignabsent𝜆subscript𝑝1𝜆𝜆subscript𝑝2subscript𝑝1subscript𝑝2\displaystyle:=\lambda-p_{1}\lambda(\lambda-p_{2}),\quad p_{1},\,p_{2}\in\mathbb{R}.

We immediately see that zero is a root of the parametric form of y1subscript𝑦1y_{1}, and the other root is p2subscript𝑝2p_{2}. Since it is exactly like the defined Linear Factors Form defined for the quadratic map in subsection 3.2 on page 3.2, we know that its maximum lies at

λc=p1p2+12p1,subscript𝜆𝑐subscript𝑝1subscript𝑝212subscript𝑝1\lambda_{c}=\frac{p_{1}\,p_{2}+1}{2\,p_{1}},

with corresponding value

y1max=λc2=(p1p2+1)24p12.superscriptsubscript𝑦1𝑚𝑎𝑥superscriptsubscript𝜆𝑐2superscriptsubscript𝑝1subscript𝑝2124superscriptsubscript𝑝12y_{1}^{max}=\lambda_{c}^{2}=\frac{(p_{1}\,p_{2}+1)^{2}}{4\,p_{1}^{2}}.

Using, for example, the values, p1=1subscript𝑝11p_{1}=1 and p2=3subscript𝑝23p_{2}=3 we know then that λc=2subscript𝜆𝑐2\lambda_{c}=2 and, therefore, y1max=4superscriptsubscript𝑦1𝑚𝑎𝑥4y_{1}^{max}=4 (this functions are shown in figure 3.3a. As for x1subscript𝑥1x_{1} it is then

x1(λ)=λ(3λ),subscript𝑥1𝜆𝜆3𝜆x_{1}(\lambda)=\lambda(3-\lambda),

where the maximum is clearly located at λ=3/2𝜆32\lambda=3/2 with a value of x1max=9/4superscriptsubscript𝑥1𝑚𝑎𝑥94x_{1}^{max}=9/4. The graph of x1subscript𝑥1x_{1} is shown in figure 3.1b. Also, x1subscript𝑥1x_{1} crosses the first bifurcation value (b1=2subscript𝑏12b_{1}=2) two times at λ=1𝜆1\lambda=1 and λ=2𝜆2\lambda=2, and it is easy to see that x1˙(λ)0˙subscript𝑥1𝜆0\dot{x_{1}}(\lambda)\neq 0 for those values of λ𝜆\lambda, so that by proposition 3.7, we expect period doubling bifurcations at this values. Also, since x1max=2.25<2.45b2superscriptsubscript𝑥1𝑚𝑎𝑥2.252.45subscript𝑏2x_{1}^{max}=2.25<2.45\approx b_{2}, according to proposition 3.11 we know that we will only get one such bifurcation leading to stable 2-cycles in the interval (1, 2)12(1,\,2), and then going back to a unique stable fixed point (i.e. x1subscript𝑥1x_{1} stays in the 1-stability band in the interval (1, 2)12(1,\,2)). Thus, this example complies with our definition of a regular-reversal map. The latter predictions are confirmed in the bifurcation diagram shown in figure 3.4.

Refer to caption
(a) Graphs of linear y0(λ)subscript𝑦0𝜆y_{0}(\lambda) and quadratic y1(λ)subscript𝑦1𝜆y_{1}(\lambda) along the selected values of λ𝜆\lambda. Also, the stability bands (b1subscript𝑏1b_{1}, b2subscript𝑏2b_{2} and bsubscript𝑏b_{\infty}) of y0subscript𝑦0y_{0} are shown around it.
Refer to caption
(b) Graph of corresponding x1(λ)subscript𝑥1𝜆x_{1}(\lambda) along with its stability bands.
Figure 3.3: Quadratic parametrization of fixed points with p1=1subscript𝑝11p_{1}=1 and p2=3subscript𝑝23p_{2}=3.
Refer to caption
Figure 3.4: Bifurcation diagram for the quadratic parametrization of the fixed points with p1=1subscript𝑝11p_{1}=1 and p2=3subscript𝑝23p_{2}=3..

Now, increasing the maximum value of x1subscript𝑥1x_{1} we can accomplish more bifurcations in the diagram. This can be achieved by increasing the slope of y1subscript𝑦1y_{1} to p1=1.1422subscript𝑝11.1422p_{1}=1.1422, since for this value x0maxbsuperscriptsubscript𝑥0𝑚𝑎𝑥subscript𝑏x_{0}^{max}\approx b_{\infty}. The corresponding graph of x1subscript𝑥1x_{1} is shown in figure 3.5a. And, also, the respective bifurcation diagram is shown in figure 3.5b, where we can see that the system undergoes a cascade of period doubling bifurcations in the central part. We thus produce another regular-reversal map with more nested regions of different types.

Refer to caption
(a) Graph of corresponding x1subscript𝑥1x_{1}. In this case x1maxbsuperscriptsubscript𝑥1𝑚𝑎𝑥subscript𝑏x_{1}^{max}\approx b_{\infty}. Also, the stability bands (b1subscript𝑏1b_{1}, b2subscript𝑏2b_{2} and bsubscript𝑏b_{\infty}) of x0subscript𝑥0x_{0} are shown above it.
Refer to caption
(b) Bifurcation diagram. We have set here x1maxbsuperscriptsubscript𝑥1𝑚𝑎𝑥subscript𝑏x_{1}^{max}\approx b_{\infty}.
Figure 3.5: Quadratic parametrization of the fixed points with p1=1.1422subscript𝑝11.1422p_{1}=1.1422 and p2=3subscript𝑝23p_{2}=3.

If we continue increasing the maximum value of x1subscript𝑥1x_{1} we can achieve a chaotic region within the regular-reversal map. Any value above the used last above will suffice. Say we take p1=1.2subscript𝑝11.2p_{1}=1.2. The graph of x1subscript𝑥1x_{1} for this case is shown in figure 3.6 and we then get the bifurcation diagram from figure 3.7, which displays a chaotic region in the central part, as stated. This is still a regular-reversal map.

Refer to caption
Figure 3.6: Graph of corresponding x1subscript𝑥1x_{1} for linear y0(λ)subscript𝑦0𝜆y_{0}(\lambda) and quadratic y1(λ)subscript𝑦1𝜆y_{1}(\lambda) with p1=1.2subscript𝑝11.2p_{1}=1.2 and p2=3subscript𝑝23p_{2}=3. The maximum value of x1(λ)subscript𝑥1𝜆x_{1}(\lambda) is set to be slightly above bsubscript𝑏b_{\infty}. Also, the bifurcation values b1subscript𝑏1b_{1}, b2subscript𝑏2b_{2} and bsubscript𝑏b_{\infty} are shown.
Refer to caption
Figure 3.7: Bifurcation diagram for the quadratic parametrization of the fixed points with p1=1.2subscript𝑝11.2p_{1}=1.2 and p2=3subscript𝑝23p_{2}=3. We have set here x1max>bsuperscriptsubscript𝑥1𝑚𝑎𝑥subscript𝑏x_{1}^{max}>b_{\infty}.

3.5.3 Sinusoidal fixed point

We can construct a periodic regular-reversal map by making one of the fixed points have a sinusoidal parametric dependence on λ𝜆\lambda. That is, we leave y0subscript𝑦0y_{0} as before and define

y1(λ):=λ+a(1+sinπλ),assignsubscript𝑦1𝜆𝜆𝑎1𝜋𝜆y_{1}(\lambda):=\lambda+a\,(1+\sin{\pi\,\lambda}), (3.32)

where we can tweak the sinusoidal form of y1subscript𝑦1y_{1} by means of the constant a𝑎a which is just an amplitude (and translation, at the same time). Choosing a=1.3b2𝑎1.3greater-than-or-equivalent-tosubscript𝑏2a=1.3\gtrsim\frac{b_{\infty}}{2} we can get a periodic regular-reversal map with chaotic regions in the periodic central regions. The graphs of the roots are shown in figure 3.8 and the bifurcation diagram is shown in figure 3.9.

Refer to caption
(a) Graphs of linear y0(λ)subscript𝑦0𝜆y_{0}(\lambda) and sinusoidal y1(λ)subscript𝑦1𝜆y_{1}(\lambda) along the selected values of λ𝜆\lambda. Also, the stability bands (b1subscript𝑏1b_{1}, b2subscript𝑏2b_{2} and bsubscript𝑏b_{\infty}) of y0subscript𝑦0y_{0} are shown around it.
Refer to caption
(b) Graph of corresponding x1(λ)subscript𝑥1𝜆x_{1}(\lambda) along with its stability bands.
Figure 3.8: Sinusoidal parametrization of fixed points.
Refer to caption
Figure 3.9: Bifurcation diagram for the sinusoidal parametrization of the fixed points example.

3.5.4 When zero loses its stability

It is important to point out that if x1subscript𝑥1x_{1} becomes negative, i.e. the difference between y1subscript𝑦1y_{1} and y0subscript𝑦0y_{0} becomes negative (assuming again M=1𝑀1M=1), then we enter the stability region of the fixed point at zero, x0subscript𝑥0x_{0}, in the canonical quadratic map (and y0subscript𝑦0y_{0} in the general quadratic map). Thus decreasing values of x1subscript𝑥1x_{1} will cause bifurcations to appear this time from the zero fixed point, as the stability bands are crossed by x1subscript𝑥1x_{1} (correspondingly, y1subscript𝑦1y_{1} in the general quadratic map). We can achieve the latter by defining again a linear y1subscript𝑦1y_{1} as

y1(λ):=p1+p2λassignsubscript𝑦1𝜆subscript𝑝1subscript𝑝2𝜆y_{1}(\lambda):=p_{1}+p_{2}\,\lambda

with, say p1=3subscript𝑝13p_{1}=3 and p2=1subscript𝑝21p_{2}=-1. Then we get y1subscript𝑦1y_{1} to cross through all the stability bands of y0subscript𝑦0y_{0} in the interval (0, 3)03(0,\,3) of the parameter λ𝜆\lambda. This is shown graphically in figure 3.10 along with its corresponding graph of x1subscript𝑥1x_{1}.

The bifurcation diagram in this case starts, from left to right, being chaotic, then reversal, then switching to stability of x0subscript𝑥0x_{0} (correspondingly, y0subscript𝑦0y_{0}) and then being regular until chaos is reached again, as x1subscript𝑥1x_{1} crosses the stability bands. This is shown in figure 3.11.

Refer to caption
(a) Graphs of linear y0(λ)subscript𝑦0𝜆y_{0}(\lambda) and linear y1(λ)subscript𝑦1𝜆y_{1}(\lambda) crossing all stability bands.
Refer to caption
(b) Graph of corresponding x1(λ)subscript𝑥1𝜆x_{1}(\lambda) crossing all stability bands.
Figure 3.10: Linear parametrization of fixed points crossing through all stability bands.
Refer to caption
Figure 3.11: Bifurcation diagram for the linear parametrization of the fixed points example that crosses through all stability bands.

3.5.5 Asymptotic fixed point

Since we know that the onset of chaos occurs at b2.57subscript𝑏2.57b_{\infty}\approx 2.57 (as calculated through approximation formula 2.6), we can force the moving x1subscript𝑥1x_{1} fixed point of the CQM to asymptotically approach this value in order to visualize more clearly the bifurcation process that takes place very near this value. Now, for symmetry reasons, it will be best if we make the x0=0subscript𝑥00x_{0}=0 fixed point to be the one that bifurcates, instead of x1subscript𝑥1x_{1}; we do this by making x1bsubscript𝑥1subscript𝑏x_{1}\rightarrow-b_{\infty} (remember (b, 0)subscript𝑏 0(-b_{\infty},\,0) is the interval where the period doubling bifurcations cascade occurs for x0=0subscript𝑥00x_{0}=0). So, we define

x1(λ):=(1)p1b+(1)p1+1exp(p2λ),assignsubscript𝑥1𝜆superscript1subscript𝑝1subscript𝑏superscript1subscript𝑝11subscript𝑝2𝜆x_{1}(\lambda):=(-1)^{p_{1}}b_{\infty}+(-1)^{p_{1}+1}\exp(-p_{2}\,\lambda), (3.33)

where p1{0, 1}subscript𝑝101p_{1}\in\{0,\,1\} and p2>0subscript𝑝20p_{2}>0. We choose for this example, for the above reasons, p1=p2=1subscript𝑝1subscript𝑝21p_{1}=p_{2}=1. The corresponding bifurcation diagram for this case is shown in figure 3.12, upper panel, where we can clearly see the bifurcations take place up to the attracting periodic point of period 26superscript262^{6} (64-cycle); the lower panel in the same figure shows the value of the fixed points, which shows the asymptotically varying x1subscript𝑥1x_{1}. The scale of the parameter value is the same in both panels to allow for comparison; notice that, when the fixed point curve crosses the marked bksubscript𝑏𝑘b_{k} values, period doubling bifurcation takes place in the bifurcation diagram, as expected. By making precise calculations based on this diagrams we could determine the bifurcation values up to an arbitrary 2Nsuperscript2𝑁2^{N}-cycle, as we did for up to N=7𝑁7N=7 in this work.

Refer to caption
Figure 3.12: Bifurcation diagram (upper) and fixed points plot with stability bands for the exponential parametrization of the fixed point example (lower). In the lower panel, we have set x1bsubscript𝑥1subscript𝑏x_{1}\rightarrow-b_{\infty} (continuous thick line), the zero fixed point is marked by a dash-dotted line and the bifurcation values b1,b2,,b5subscript𝑏1subscript𝑏2subscript𝑏5-b_{1},\,-b_{2},...,\,-b_{5} are also shown as dashed lines.

Chapter 4 Cubic maps

As stated before, in chapter 3 we furthered the results of chapter 2, mainly based on the work of [Solís and Jódar(2004)], and restated them in the frame of a new formulation that would allow for generalization. Well, the first such generalization comes precisely for third degree polynomial maps, i.e. cubic polynomials.

4.1 General Cubic Map

Consider a cubic map (iteration function) in its general form, as stated by the

Definition 4.1 (General Cubic Map).

The General Cubic Map is defined by

f3(y):=y+Pf3(y),assignsubscript𝑓3𝑦𝑦subscript𝑃subscript𝑓3𝑦f_{3}(y):=y+P_{f_{3}}(y), (4.1)

where

Pf3(y)=α+βy+γy2+δy3subscript𝑃subscript𝑓3𝑦𝛼𝛽𝑦𝛾superscript𝑦2𝛿superscript𝑦3P_{f_{3}}(y)=\alpha+\beta\,y+\gamma\,y^{2}+\delta\,y^{3} (4.2)

is called the Fixed Points Polynomial of f3subscript𝑓3f_{3}. All the coefficients α,β,γ𝛼𝛽𝛾\alpha,\,\beta,\,\gamma and δ𝛿\delta are functions of the parameter λ𝜆\lambda.

It is evident that any cubic map can be put in this form by adjusting the corresponding values of the coefficients in the fixed points polynomial. By the fundamental theorem of algebra, we know that (4.2) has three roots, by which the GCM has three fixed points. The roots of Pf3subscript𝑃subscript𝑓3P_{f_{3}} are then [Spiegel et al.(2009)Spiegel, Lipschutz, and Liu]

y0subscript𝑦0\displaystyle y_{0} =S+T13γ~absent𝑆𝑇13~𝛾\displaystyle=S+T-\frac{1}{3}\tilde{\gamma} (4.3)
y1subscript𝑦1\displaystyle y_{1} =12(S+T)13γ~+12i3(ST)absent12𝑆𝑇13~𝛾12𝑖3𝑆𝑇\displaystyle=-\frac{1}{2}(S+T)-\frac{1}{3}\tilde{\gamma}+\frac{1}{2}i\,\sqrt{3}(S-T)
y2subscript𝑦2\displaystyle y_{2} =12(S+T)13γ~12i3(ST)absent12𝑆𝑇13~𝛾12𝑖3𝑆𝑇\displaystyle=-\frac{1}{2}(S+T)-\frac{1}{3}\tilde{\gamma}-\frac{1}{2}i\,\sqrt{3}(S-T)

where α~=α/δ~𝛼𝛼𝛿\tilde{\alpha}=\alpha/\delta, β~=β/δ~𝛽𝛽𝛿\tilde{\beta}=\beta/\delta, γ~=γ/δ~𝛾𝛾𝛿\tilde{\gamma}=\gamma/\delta, and

Q𝑄\displaystyle Q =3β~γ~29,absent3~𝛽superscript~𝛾29\displaystyle=\frac{3\tilde{\beta}-\tilde{\gamma}^{2}}{9}, R𝑅\displaystyle R =9β~γ~27α~2γ~354,absent9~𝛽~𝛾27~𝛼2superscript~𝛾354\displaystyle=\frac{9\tilde{\beta}\tilde{\gamma}-27\tilde{\alpha}-2\tilde{\gamma}^{3}}{54}, (4.4)
D𝐷\displaystyle D =Q3+R2,absentsuperscript𝑄3superscript𝑅2\displaystyle=Q^{3}+R^{2},
S𝑆\displaystyle S =R+D3,absent3𝑅𝐷\displaystyle=\sqrt[3]{R+\sqrt{D}}, T𝑇\displaystyle T =RD3.absent3𝑅𝐷\displaystyle=\sqrt[3]{R-\sqrt{D}}.

The coefficients of eq. (4.2) and its fixed points are related by

y0+y1+y2subscript𝑦0subscript𝑦1subscript𝑦2\displaystyle y_{0}+y_{1}+y_{2} =γ~,absent~𝛾\displaystyle=-\tilde{\gamma}, y0y1+y1y2+y2y0subscript𝑦0subscript𝑦1subscript𝑦1subscript𝑦2subscript𝑦2subscript𝑦0\displaystyle y_{0}y_{1}+y_{1}y_{2}+y_{2}y_{0} =β~,absent~𝛽\displaystyle=\tilde{\beta}, y0y1y2subscript𝑦0subscript𝑦1subscript𝑦2\displaystyle y_{0}y_{1}y_{2} =α~.absent~𝛼\displaystyle=-\tilde{\alpha}. (4.5)

D𝐷D is called the discriminant and we have three cases:

  • If D>0𝐷0D>0 then one fixed point is real and the other two are complex conjugates.

  • If D=0𝐷0D=0 then the three fixed points are real with at least two of them equal.

  • If D<0𝐷0D<0 then all fixed points are real and distinct.

It is the last two cases (real fixed points) that will interest us most for the time being. Suppose in particular that D<0𝐷0D<0. Then, we can write [Spiegel et al.(2009)Spiegel, Lipschutz, and Liu]

y0subscript𝑦0\displaystyle y_{0} =2Qcos(θ3)13β~,absent2𝑄𝜃313~𝛽\displaystyle=2\sqrt{-Q}\,\cos\left(\frac{\theta}{3}\right)-\frac{1}{3}\tilde{\beta}, (4.6)
y1subscript𝑦1\displaystyle y_{1} =2Qcos(θ+π3)13β~,absent2𝑄𝜃𝜋313~𝛽\displaystyle=2\sqrt{-Q}\,\cos\left(\frac{\theta+\pi}{3}\right)-\frac{1}{3}\tilde{\beta},
y2subscript𝑦2\displaystyle y_{2} =2Qcos(θ+2π3)13β~,absent2𝑄𝜃2𝜋313~𝛽\displaystyle=2\sqrt{-Q}\,\cos\left(\frac{\theta+2\pi}{3}\right)-\frac{1}{3}\tilde{\beta},

where cosθ=R/Q3𝜃𝑅superscript𝑄3\cos\theta=R/\sqrt{-Q^{3}}.

4.2 Linear Factors Product Form

Since we know that (4.2) has three roots, by the factor theorem we can rewrite Pf3subscript𝑃subscript𝑓3P_{f_{3}} as

Pf3(y)subscript𝑃subscript𝑓3𝑦\displaystyle P_{f_{3}}(y) =M(yy0)(yy1)(yy2),absent𝑀𝑦subscript𝑦0𝑦subscript𝑦1𝑦subscript𝑦2\displaystyle=M(y-y_{0})(y-y_{1})(y-y_{2}), (4.7)
=sM~(yy0)(yy1)(yy2)absent𝑠~𝑀𝑦subscript𝑦0𝑦subscript𝑦1𝑦subscript𝑦2\displaystyle=s\tilde{M}(y-y_{0})(y-y_{1})(y-y_{2})

where s=sign(M)𝑠sign𝑀s=\mathrm{sign}(M), M~=|M|~𝑀𝑀\tilde{M}=|M|; and then rewrite (4.1) as

Definition 4.2 (Linear Factors Form of the Cubic Map).

Let f3subscript𝑓3f_{3} be a general cubic map with three fixed points, y0subscript𝑦0y_{0}, y1subscript𝑦1y_{1} and y2subscript𝑦2y_{2}\in\mathbb{C}. We can write f3subscript𝑓3f_{3} as

h3(y)subscript3𝑦\displaystyle h_{3}(y) =y+M(yy0)(yy1)(yy2)absent𝑦𝑀𝑦subscript𝑦0𝑦subscript𝑦1𝑦subscript𝑦2\displaystyle=y+M(y-y_{0})(y-y_{1})(y-y_{2}) (4.8)
=y+sM~(yy0)(yy1)(yy2).absent𝑦𝑠~𝑀𝑦subscript𝑦0𝑦subscript𝑦1𝑦subscript𝑦2\displaystyle=y+s\tilde{M}(y-y_{0})(y-y_{1})(y-y_{2}).

where all M,y0,y1𝑀subscript𝑦0subscript𝑦1M,\,y_{0},\,y_{1} and y2subscript𝑦2y_{2} are functions of the parameter λ𝜆\lambda. We call h3subscript3h_{3} the Linear Factors Form of the cubic map (LFFCM).

The LFFCM has the derivatives

h(y)superscript𝑦\displaystyle h^{\prime}(y) =1+M[(yy1)(yy2)+(yy0)(yy2)+(yy0)(yy1)],absent1𝑀delimited-[]𝑦subscript𝑦1𝑦subscript𝑦2𝑦subscript𝑦0𝑦subscript𝑦2𝑦subscript𝑦0𝑦subscript𝑦1\displaystyle=1+M\left[(y-y_{1})(y-y_{2})+(y-y_{0})(y-y_{2})+(y-y_{0})(y-y_{1})\right], (4.9)
h′′(y)superscript′′𝑦\displaystyle h^{\prime\prime}(y) =2M(y0+y1+y23y).absent2𝑀subscript𝑦0subscript𝑦1subscript𝑦23𝑦\displaystyle=-2\,M\,\left(y_{0}+y_{1}+y_{2}-3\,y\right).

By equating the first derivative to zero we can obtain the two critical points

y1csuperscriptsubscript𝑦1𝑐\displaystyle y_{1}^{c} =13[(y0+y1+y2)+12[(y1y0)2+(y2y1)2+(y2y0)2]3/M],absent13delimited-[]subscript𝑦0subscript𝑦1subscript𝑦212delimited-[]superscriptsubscript𝑦1subscript𝑦02superscriptsubscript𝑦2subscript𝑦12superscriptsubscript𝑦2subscript𝑦023𝑀\displaystyle=\frac{1}{3}\left[(y_{0}+y_{1}+y_{2})+\sqrt{\frac{1}{2}\left[(y_{1}-y_{0})^{2}+(y_{2}-y_{1})^{2}+(y_{2}-y_{0})^{2}\right]-3/M}\right], (4.10)
y1csuperscriptsubscript𝑦1𝑐\displaystyle y_{1}^{c} =13[(y0+y1+y2)12[(y1y0)2+(y2y1)2+(y2y0)2]3/M]absent13delimited-[]subscript𝑦0subscript𝑦1subscript𝑦212delimited-[]superscriptsubscript𝑦1subscript𝑦02superscriptsubscript𝑦2subscript𝑦12superscriptsubscript𝑦2subscript𝑦023𝑀\displaystyle=\frac{1}{3}\left[(y_{0}+y_{1}+y_{2})-\sqrt{\frac{1}{2}\left[(y_{1}-y_{0})^{2}+(y_{2}-y_{1})^{2}+(y_{2}-y_{0})^{2}\right]-3/M}\right]

and, evaluating these critical points in h′′superscript′′h^{\prime\prime} we have

h′′(y1c)superscript′′superscriptsubscript𝑦1𝑐\displaystyle h^{\prime\prime}(y_{1}^{c}) =2M12[(y1y0)2+(y2y1)2+(y2y0)2]3/M,absent2𝑀12delimited-[]superscriptsubscript𝑦1subscript𝑦02superscriptsubscript𝑦2subscript𝑦12superscriptsubscript𝑦2subscript𝑦023𝑀\displaystyle=2M\sqrt{\frac{1}{2}\left[(y_{1}-y_{0})^{2}+(y_{2}-y_{1})^{2}+(y_{2}-y_{0})^{2}\right]-3/M}, (4.11)
h′′(y2c)superscript′′superscriptsubscript𝑦2𝑐\displaystyle h^{\prime\prime}(y_{2}^{c}) =2M12[(y1y0)2+(y2y1)2+(y2y0)2]3/M=h′′(y1c),absent2𝑀12delimited-[]superscriptsubscript𝑦1subscript𝑦02superscriptsubscript𝑦2subscript𝑦12superscriptsubscript𝑦2subscript𝑦023𝑀superscript′′superscriptsubscript𝑦1𝑐\displaystyle=-2M\sqrt{\frac{1}{2}\left[(y_{1}-y_{0})^{2}+(y_{2}-y_{1})^{2}+(y_{2}-y_{0})^{2}\right]-3/M}=-h^{\prime\prime}(y_{1}^{c}),

so that necessarily one critical point is a maximum and the other one is a minimum, unless they are both equal, in which case we have a saddle point.

4.3 Canonical Cubic Map

As in the case for the Canonical Quadratic Map, we will apply a linear transformation to (4.8) so that one fixed point is mapped to zero and the “amplitude” of the linear factors term is unity. We know that, since f3subscript𝑓3f_{3} is cubic, at least one fixed point is real, so we can map this fixed point to zero. This can be accomplished by one of the following transformations

T0(x)subscript𝑇0𝑥\displaystyle T_{0}(x) =y0±sM~1/2x,absentplus-or-minussubscript𝑦0𝑠superscript~𝑀12𝑥\displaystyle=y_{0}\pm s\tilde{M}^{-1/2}x, T1(x)subscript𝑇1𝑥\displaystyle T_{1}(x) =y1±sM~1/2x,absentplus-or-minussubscript𝑦1𝑠superscript~𝑀12𝑥\displaystyle=y_{1}\pm s\tilde{M}^{-1/2}x, T2(x)subscript𝑇2𝑥\displaystyle T_{2}(x) =y2±sM~1/2x,absentplus-or-minussubscript𝑦2𝑠superscript~𝑀12𝑥\displaystyle=y_{2}\pm s\tilde{M}^{-1/2}x, (4.12)

by taking y=Tk(x),k{0, 1, 2}formulae-sequence𝑦subscript𝑇𝑘𝑥𝑘012y=T_{k}(x),\,k\in\{0,\,1,\,2\} if yksubscript𝑦𝑘y_{k} is real. Without loss of generality, we will assume y0subscript𝑦0y_{0} is real and apply T0subscript𝑇0T_{0} with the plus sign calling it simply T𝑇T, so that, once the calculations are done, we get

Definition 4.3 (Canonical Cubic Map).

The Canonical Cubic Map (CCM) is defined by

g3(x;λ)=x+sx(xx1(λ))(xx2(λ)),subscript𝑔3𝑥𝜆𝑥𝑠𝑥𝑥subscript𝑥1𝜆𝑥subscript𝑥2𝜆g_{3}(x;\lambda)=x+sx(x-x_{1}(\lambda))(x-x_{2}(\lambda)), (4.13)

where it has been stressed out that both fixed points x1subscript𝑥1x_{1} and x2subscript𝑥2x_{2} depend upon the parameter λ𝜆\lambda.

So that, if M>0𝑀0M>0 then

g3(x)=x+x(xx1)(xx2),subscript𝑔3𝑥𝑥𝑥𝑥subscript𝑥1𝑥subscript𝑥2g_{3}(x)=x+x(x-x_{1})(x-x_{2}), (4.14)

and if M<0𝑀0M<0

g3(x)=xx(xx1)(xx2),subscript𝑔3𝑥𝑥𝑥𝑥subscript𝑥1𝑥subscript𝑥2g_{3}(x)=x-x(x-x_{1})(x-x_{2}), (4.15)

From the easily verifiable calculations, it can be shown that the relation between the roots of the linear factors form of the cubic map and the canonical cubic map is given by

Lemma 4.1.

The fixed points of the linear factors form of the cubic map and the canonical cubic map are related by

x1(λ)subscript𝑥1𝜆\displaystyle x_{1}(\lambda) =sM~1/2[y1(λ)y0(λ)],absent𝑠superscript~𝑀12delimited-[]subscript𝑦1𝜆subscript𝑦0𝜆\displaystyle=s\tilde{M}^{1/2}\left[y_{1}(\lambda)-y_{0}(\lambda)\right], x2(λ)subscript𝑥2𝜆\displaystyle x_{2}(\lambda) =sM~1/2[y2(λ)y0(λ)].absent𝑠superscript~𝑀12delimited-[]subscript𝑦2𝜆subscript𝑦0𝜆\displaystyle=s\tilde{M}^{1/2}\left[y_{2}(\lambda)-y_{0}(\lambda)\right]. (4.16)

We have then again reduced the parametric dependence to only two functions of the parameter λ𝜆\lambda: x1subscript𝑥1x_{1} and x2subscript𝑥2x_{2}. Notice T𝑇T is a homeomorphism between the domains of both maps; this will help us in chapter 5 to prove that the Linear Factors Form and the Canonical Form of polynomial maps are actually topologically conjugate, which in turn means that the stability and chaos properties are preserved between the maps, which allows us to determine stability properties for any cubic map by analyzing only the CCM.

Now, one first immediate result is

Proposition 4.2.

The Canonical Cubic Map has three roots:

x0rsuperscriptsubscript𝑥0𝑟\displaystyle x_{0}^{r} =0,absent0\displaystyle=0, x1rsuperscriptsubscript𝑥1𝑟\displaystyle x_{1}^{r} =x1+x2+s(x2x1)24s2,absentsubscript𝑥1subscript𝑥2𝑠superscriptsubscript𝑥2subscript𝑥124𝑠2\displaystyle=\frac{x_{1}+x_{2}+s\sqrt{(x_{2}-x_{1})^{2}-4s}}{2}, x2r=x1+x2s(x2x1)24s2.superscriptsubscript𝑥2𝑟subscript𝑥1subscript𝑥2𝑠superscriptsubscript𝑥2subscript𝑥124𝑠2\displaystyle x_{2}^{r}=\frac{x_{1}+x_{2}-s\sqrt{(x_{2}-x_{1})^{2}-4s}}{2}. (4.17)

Moreover, if M>0𝑀0M>0

  • If x2x1subscript𝑥2subscript𝑥1x_{2}\neq-x_{1} and x2x1=±2subscript𝑥2subscript𝑥1plus-or-minus2x_{2}-x_{1}=\pm 2, we will have a single root different from zero with multiplicity of two, and equal to (x1+x2)/2subscript𝑥1subscript𝑥22(x_{1}+x_{2})/2, i.e. the midpoint between the two non-zero fixed points.

  • If x2=x1subscript𝑥2subscript𝑥1x_{2}=-x_{1} and x2x1=±2subscript𝑥2subscript𝑥1plus-or-minus2x_{2}-x_{1}=\pm 2, zero will be the only root, with multiplicity of three.

  • If |x2x1|>2subscript𝑥2subscript𝑥12|x_{2}-x_{1}|>2, we will have two real and distinct non-zero roots.

  • If |x2x1|<2subscript𝑥2subscript𝑥12|x_{2}-x_{1}|<2, zero will be the only real root of the CCM and the other two roots will be complex conjugates.

And if M<0𝑀0M<0 the three roots are always real and distinct, except when x1=x2=±1subscript𝑥1subscript𝑥2plus-or-minus1x_{1}=x_{2}=\pm 1 when x0r=0superscriptsubscript𝑥0𝑟0x_{0}^{r}=0 has multiplicity of two.

As for the derivatives of the CCM, they are

g3(x)superscriptsubscript𝑔3𝑥\displaystyle g_{3}^{\prime}(x) =1+s[(xx1)(xx2)+x(xx2)+x(xx1)],absent1𝑠delimited-[]𝑥subscript𝑥1𝑥subscript𝑥2𝑥𝑥subscript𝑥2𝑥𝑥subscript𝑥1\displaystyle=1+s\left[(x-x_{1})(x-x_{2})+x(x-x_{2})+x(x-x_{1})\right], (4.18)
g3′′(x)superscriptsubscript𝑔3′′𝑥\displaystyle g_{3}^{\prime\prime}(x) =s[6x2(x1+x2)]absent𝑠delimited-[]6𝑥2subscript𝑥1subscript𝑥2\displaystyle=s\left[6x-2(x_{1}+x_{2})\right]

from which we can determine that the critical points are

x1csuperscriptsubscript𝑥1𝑐\displaystyle x_{1}^{c} =13[x1+x2+sx12x1x2+x223s],absent13delimited-[]subscript𝑥1subscript𝑥2𝑠superscriptsubscript𝑥12subscript𝑥1subscript𝑥2superscriptsubscript𝑥223𝑠\displaystyle=\frac{1}{3}\left[x_{1}+x_{2}+s\sqrt{x_{1}^{2}-x_{1}x_{2}+x_{2}^{2}-3s}\right], (4.19)
x2csuperscriptsubscript𝑥2𝑐\displaystyle x_{2}^{c} =13[x1+x2sx12x1x2+x223s].absent13delimited-[]subscript𝑥1subscript𝑥2𝑠superscriptsubscript𝑥12subscript𝑥1subscript𝑥2superscriptsubscript𝑥223𝑠\displaystyle=\frac{1}{3}\left[x_{1}+x_{2}-s\sqrt{x_{1}^{2}-x_{1}x_{2}+x_{2}^{2}-3s}\right].

So that, evaluating g′′superscript𝑔′′g^{\prime\prime} at the critical values of g,𝑔g, x1csuperscriptsubscript𝑥1𝑐x_{1}^{c} and x2csuperscriptsubscript𝑥2𝑐x_{2}^{c} we have thatwe have that

g3′′(x1c)=g3′′(x2c)=2x12x1x2+x223s,superscriptsubscript𝑔3′′superscriptsubscript𝑥1𝑐superscriptsubscript𝑔3′′superscriptsubscript𝑥2𝑐2superscriptsubscript𝑥12subscript𝑥1subscript𝑥2superscriptsubscript𝑥223𝑠g_{3}^{\prime\prime}(x_{1}^{c})=-g_{3}^{\prime\prime}(x_{2}^{c})=2\sqrt{x_{1}^{2}-x_{1}x_{2}+x_{2}^{2}-3s},

and, then, we have as expected one maximum and one minimum unless both are equal, in which case we have a saddle point. From this we have proved

Proposition 4.3.

Consider the CCM as defined above. Then, if M>0𝑀0M>0

  • If x12+x22>3+x1x2superscriptsubscript𝑥12superscriptsubscript𝑥223subscript𝑥1subscript𝑥2x_{1}^{2}+x_{2}^{2}>3+x_{1}x_{2}, both critical points are real and distinct, one corresponding to a maximum and the other to a minimum.

  • If x12+x22=3+x1x2superscriptsubscript𝑥12superscriptsubscript𝑥223subscript𝑥1subscript𝑥2x_{1}^{2}+x_{2}^{2}=3+x_{1}x_{2}, both critical points are equal and correspond to a saddle point.

  • If x12+x22<3+x1x2superscriptsubscript𝑥12superscriptsubscript𝑥223subscript𝑥1subscript𝑥2x_{1}^{2}+x_{2}^{2}<3+x_{1}x_{2}, the two critical points are complex conjugates, so we do not have any real critical points.

And if M<0𝑀0M<0,

  • If x12+x22+3>x1x2superscriptsubscript𝑥12superscriptsubscript𝑥223subscript𝑥1subscript𝑥2x_{1}^{2}+x_{2}^{2}+3>x_{1}x_{2}, both critical points are real and distinct, one corresponding to a maximum and the other to a minimum.

  • If x12+x22+3=x1x2superscriptsubscript𝑥12superscriptsubscript𝑥223subscript𝑥1subscript𝑥2x_{1}^{2}+x_{2}^{2}+3=x_{1}x_{2}, both critical points are equal and correspond to a saddle point.

  • If x12+x22+3<x1x2superscriptsubscript𝑥12superscriptsubscript𝑥223subscript𝑥1subscript𝑥2x_{1}^{2}+x_{2}^{2}+3<x_{1}x_{2}, the two critical points are complex conjugates, so we do not have any (real) critical points.

4.4 Stability and Chaos in the Canonical Cubic Map

Next, let us determine the stability of the periodic points of the CCM. As we will see further below in chapter 5, this analysis will suffice for any cubic map, by means of topological conjugacy. However, we can only explicitly give this for the fixed points.

4.4.1 Fixed Points

We already know, by construction, that the fixed points of the CCM are x0=0,x1subscript𝑥00subscript𝑥1x_{0}=0,\,x_{1} and x2subscript𝑥2x_{2}. While the first is constant always, the other two fixed points are set to be functions of the parameter λ𝜆\lambda. By evaluating in g3superscriptsubscript𝑔3g_{3}^{\prime} we get the eigenvalue functions. For x0=0subscript𝑥00x_{0}=0 we have

ϕ0(λ)=g3(0)=sx1(λ)x2(λ)+1,subscriptitalic-ϕ0𝜆superscriptsubscript𝑔30𝑠subscript𝑥1𝜆subscript𝑥2𝜆1\phi_{0}(\lambda)=g_{3}^{\prime}(0)=sx_{1}(\lambda)x_{2}(\lambda)+1,

So that, the stability condition for this fixed point is

2<sx1x2<0.2𝑠subscript𝑥1subscript𝑥20-2<sx_{1}x_{2}<0. (4.20)

We can draw some conclusions from this. In order for zero to be a stable (attracting) fixed point one must have:

Lemma 4.4.

The following are sufficient conditions for the asymptotic stability of the zero fixed point of the Canonical Cubic Map:

  • in magnitude, |x1||x2|<2subscript𝑥1subscript𝑥22|x_{1}||x_{2}|<2; and

  • if M>0𝑀0M>0, x1subscript𝑥1x_{1} and x2subscript𝑥2x_{2} must have different signs; or

  • if M<0𝑀0M<0, x1subscript𝑥1x_{1} and x2subscript𝑥2x_{2} must have the same sign.

Notice that the stability condition (4.20) states that the product of the distances from the other two fixed points to the zero fixed point must within the range (2, 0)2 0(-2,\,0), for positive M𝑀M [or (0, 2)02(0,\,2) for negative M𝑀M], for the zero fixed point to be asymptotically stable.

The case of xk=0subscript𝑥𝑘0x_{k}=0, k{1, 2}𝑘12k\in\{1,\,2\}, is not included in the discussion here since this would represent repeated fixed points (multiplicity), which will be discussed in section 4.6 below; likewise, in the remainder of this section we will avoid dealing with multiplicity of the fixed points.

Now, for x1subscript𝑥1x_{1}, its eigenvalue function is

ϕ1(λ)=g3(x1(λ))=1+sx1(λ)(x1(λ)x2(λ)),subscriptitalic-ϕ1𝜆superscriptsubscript𝑔3subscript𝑥1𝜆1𝑠subscript𝑥1𝜆subscript𝑥1𝜆subscript𝑥2𝜆\phi_{1}(\lambda)=g_{3}^{\prime}(x_{1}(\lambda))=1+sx_{1}(\lambda)(x_{1}(\lambda)-x_{2}(\lambda)),

so that the stability condition for this fixed point is

2<sx1(x1x2)<0.2𝑠subscript𝑥1subscript𝑥1subscript𝑥20-2<sx_{1}(x_{1}-x_{2})<0. (4.21)

This gives us the following

Lemma 4.5.

The following are sufficient conditions for the asymptotic stability of the x1subscript𝑥1x_{1} fixed point:

If M>0𝑀0M>0 then

  • x1subscript𝑥1x_{1} and x2subscript𝑥2x_{2} must have the same sign; and

  • |x1|<|x2|<|x1+2/x1|subscript𝑥1subscript𝑥2subscript𝑥12subscript𝑥1|x_{1}|<|x_{2}|<|x_{1}+2/x_{1}|.

On the other hand, if M<0𝑀0M<0,

  • |x2|<|x1|subscript𝑥2subscript𝑥1|x_{2}|<|x_{1}|; and

  • if |x1|2subscript𝑥12|x_{1}|\geq\sqrt{2}, then |x12/x1|<|x2|<|x1|subscript𝑥12subscript𝑥1subscript𝑥2subscript𝑥1|x_{1}-2/x_{1}|<|x_{2}|<|x_{1}|; or

  • if 0<x1<20subscript𝑥120<x_{1}<\sqrt{2}, then x12/x1<x2<x1subscript𝑥12subscript𝑥1subscript𝑥2subscript𝑥1x_{1}-2/x_{1}<x_{2}<x_{1}; or

  • if 2<x1<02subscript𝑥10-\sqrt{2}<x_{1}<0, then x1<x2<x12/x1subscript𝑥1subscript𝑥2subscript𝑥12subscript𝑥1x_{1}<x_{2}<x_{1}-2/x_{1}.

Again, notice that the stability condition (4.21) for x1subscript𝑥1x_{1} can be translated as that the product of the distances between the other two fixed points and x1subscript𝑥1x_{1} must be within the range (2, 0)2 0(-2,\,0) for positive M𝑀M [or (0, 2)02(0,\,2) for negative M𝑀M]. Also notice that when 0<|x1|<20subscript𝑥120<|x_{1}|<\sqrt{2} the bound x12/x1subscript𝑥12subscript𝑥1x_{1}-2/x_{1} may be negative even if x1>0subscript𝑥10x_{1}>0 or positive even if x1<0subscript𝑥10x_{1}<0, therefore the usefulness of the distinction.

Finally, for x2subscript𝑥2x_{2} we have the eigenvalue function

ϕ2(λ)=g3(x2(λ))=1+sx2(λ)(x2(λ)x1(λ)),subscriptitalic-ϕ2𝜆superscriptsubscript𝑔3subscript𝑥2𝜆1𝑠subscript𝑥2𝜆subscript𝑥2𝜆subscript𝑥1𝜆\phi_{2}(\lambda)=g_{3}^{\prime}(x_{2}(\lambda))=1+sx_{2}(\lambda)(x_{2}(\lambda)-x_{1}(\lambda)),

which gives the stability condition

2<sx2(x2x1)<0,2𝑠subscript𝑥2subscript𝑥2subscript𝑥10-2<sx_{2}(x_{2}-x_{1})<0, (4.22)

which in turn produces the following

Lemma 4.6.

The following are sufficient conditions for the asymptotic stability of the x2subscript𝑥2x_{2} fixed point of the Canonical Cubic Map:

If M>0𝑀0M>0 then

  • x1subscript𝑥1x_{1} and x2subscript𝑥2x_{2} must have the same sign; and

  • |x2|<|x1|<|x2+2/x2|subscript𝑥2subscript𝑥1subscript𝑥22subscript𝑥2|x_{2}|<|x_{1}|<|x_{2}+2/x_{2}|.

On the other hand, if M<0𝑀0M<0,

  • |x1|<|x2|subscript𝑥1subscript𝑥2|x_{1}|<|x_{2}|; and

  • if |x2|2subscript𝑥22|x_{2}|\geq\sqrt{2}, then |x22/x2|<|x1|<|x2|subscript𝑥22subscript𝑥2subscript𝑥1subscript𝑥2|x_{2}-2/x_{2}|<|x_{1}|<|x_{2}|; or

  • if 0<x2<20subscript𝑥220<x_{2}<\sqrt{2}, then x22/x2<x1<x2subscript𝑥22subscript𝑥2subscript𝑥1subscript𝑥2x_{2}-2/x_{2}<x_{1}<x_{2}; or

  • if 2<x2<02subscript𝑥20-\sqrt{2}<x_{2}<0, then x2<x1<x22/x2subscript𝑥2subscript𝑥1subscript𝑥22subscript𝑥2x_{2}<x_{1}<x_{2}-2/x_{2}.

Likewise, we see that the stability condition (4.22) involves the product of the distances from the other two fixed points to the one whose stability is of interest. We will later generalize these “stability conditions” to functions of the parameter which are different for each fixed point, but of whose value depend the stability of not only the fixed points, but higher period periodic points also, through period doubling bifurcations. From the stability conditions for the three fixed points we have proved the following

Corollary.

A cubic polynomial map with three different real roots can only have a single attracting fixed point.

Proof.

Compare the stability conditions for the three fixed points. ∎

And,

Theorem 4.7.

If M>0𝑀0M>0, then sufficient conditions for the stability of a fixed point of the canonical cubic map are

  • The product of the distances between each unstable fixed point and the stable one must be negative, which means one distance is positive and the other negative, which leads us to

  • the fixed point that lies between the other two will be stable, while the outer fixed points will be unstable, as long as

  • the product of the distances between each unstable fixed point and the stable one must be greater than -2.

4.5 Higher period periodic points

Although, as previously stated, in general, we cannot calculate the values of the periodic points of period 2 or higher, we can calculate for which values of the stability conditions above the fixed points undergo period doubling bifurcations. We will see in chapter 5 that these stability conditions can actually be generalized to something called the “Product Distance Function”, which depends on the parameter and is different for each fixed point. Example 4.4 below allowed us to determine the bifurcation values, cksubscript𝑐𝑘c_{k}, of the fixed points of the CCM up to some precision. The values obtained are shown in table 4.1. When the stability conditions of each fixed point crosses these values, bifurcations take place.

k𝑘k cksubscript𝑐𝑘c_{k}
0 0
1 2
2 3.0±0.005plus-or-minus3.00.0053.0\pm 0.005
3 3.236±0.002plus-or-minus3.2360.0023.236\pm 0.002
4 3.288±0.0005plus-or-minus3.2880.00053.288\pm 0.0005
5 3.29925±0.00025plus-or-minus3.299250.000253.29925\pm 0.00025
\vdots \vdots
\infty 3.30228±5×106similar-toabsentplus-or-minus3.302285superscript106\sim 3.30228\pm 5\times 10^{-6}
Table 4.1: Bifurcation values for the Canonical Cubic Map. c0=0subscript𝑐00c_{0}=0 is included only as a reference, although it is not a bifurcation value.

From these values, we can construct the analogue of the stability bands of the CQM for the CCM.

Definition 4.4 (Stability Bands of the CCM).

Let x1,x2:𝒜:subscript𝑥1subscript𝑥2𝒜x_{1},\,x_{2}:\mathcal{A}\subseteq\mathbb{R}\rightarrow\mathbb{R} be the two nonzero fixed points of the family of cubic maps g3subscript𝑔3g_{3}, as given by definition 4.3, and {ck}ksubscriptsubscript𝑐𝑘𝑘\{c_{k}\}_{k\in\mathbb{N}} the sequence of bifurcation values of table 4.1. The open interval

(ck+1,ck),λ𝒜subscript𝑐𝑘1subscript𝑐𝑘𝜆𝒜\left(-c_{k+1},\,-c_{k}\right),\quad\lambda\in\mathcal{A} (4.23)

is called the k𝑘k-th stability band of the CCM.

Notice, however, that in contrast with the stability bands of the CQM, the stability bands of the CCM cannot be plotted along the fixed points plots, at least not directly as just defined, but rather they must be represented in a separate plot for the stability conditions, as we shall see in the examples of section 4.7 further below.

4.6 Multiplicity of the fixed points

When multiplicity of the fixed points takes place in the CCM, without loss of generality, g3subscript𝑔3g_{3} can take the following forms

g3(x)={x+sx2(xx1),if x2=x0=0,x10x+sx(xx1)2,if x1=x20x+sx3,if x1=x2=0.subscript𝑔3𝑥cases𝑥𝑠superscript𝑥2𝑥subscript𝑥1formulae-sequenceif subscript𝑥2subscript𝑥00subscript𝑥10𝑥𝑠𝑥superscript𝑥subscript𝑥12if subscript𝑥1subscript𝑥20𝑥𝑠superscript𝑥3if subscript𝑥1subscript𝑥20g_{3}(x)=\begin{cases}x+sx^{2}(x-x_{1}),&\text{if }x_{2}=x_{0}=0,\,x_{1}\neq 0\\ x+sx(x-x_{1})^{2},&\text{if }x_{1}=x_{2}\neq 0\\ x+sx^{3},&\text{if }x_{1}=x_{2}=0.\end{cases} (4.24)

with corresponding derivatives

g3(x)={1+2sx(xx1)+sx2,if x2=x0=0,x101+sx(xx1)2+2sx(xx1),if x1=x201+3sx2,if x1=x2=0.subscriptsuperscript𝑔3𝑥cases12𝑠𝑥𝑥subscript𝑥1𝑠superscript𝑥2formulae-sequenceif subscript𝑥2subscript𝑥00subscript𝑥101𝑠𝑥superscript𝑥subscript𝑥122𝑠𝑥𝑥subscript𝑥1if subscript𝑥1subscript𝑥2013𝑠superscript𝑥2if subscript𝑥1subscript𝑥20g^{\prime}_{3}(x)=\begin{cases}1+2sx(x-x_{1})+sx^{2},&\text{if }x_{2}=x_{0}=0,\,x_{1}\neq 0\\ 1+sx(x-x_{1})^{2}+2sx(x-x_{1}),&\text{if }x_{1}=x_{2}\neq 0\\ 1+3sx^{2},&\text{if }x_{1}=x_{2}=0.\end{cases} (4.25)

and therefore, g3(xk)=1,k{0, 1, 2}formulae-sequencesubscriptsuperscript𝑔3subscript𝑥𝑘1𝑘012g^{\prime}_{3}(x_{k})=1,\,k\in\{0,\,1,\,2\}, for all three cases, so that we deal with nonhyperbolic fixed points, according to definition 1.6.

Proposition 4.8.

The stability of the fixed points of the CCM when they present multiplicity is, without loss of generality, as follows

  1. 1.

    If x2=x0=0,x10formulae-sequencesubscript𝑥2subscript𝑥00subscript𝑥10x_{2}=x_{0}=0,\,x_{1}\neq 0, the zero fixed point is an unstable fixed point with multiplicity of two; and

    • if M>0𝑀0M>0 and

      • if x1>0subscript𝑥10x_{1}>0 it is semiasymptotically stable from the right;

      • if x1<0subscript𝑥10x_{1}<0 it is semiasymptotically stable from the left;

    • or if M<0𝑀0M<0 and

      • if x1>0subscript𝑥10x_{1}>0 it is semiasymptotically stable from the left;

      • if x1<0subscript𝑥10x_{1}<0 it is semiasymptotically stable from the right.

  2. 2.

    If x1=x20subscript𝑥1subscript𝑥20x_{1}=x_{2}\neq 0, this fixed point has multiplicity of two and it is unstable; moreover

    • if M>0𝑀0M>0 and

      • if x1>0subscript𝑥10x_{1}>0 it is semiasymptotically stable from the left;

      • if x1<0subscript𝑥10x_{1}<0 it is semiasymptotically stable from the right;

    • or if M<0𝑀0M<0 and

      • if x1>0subscript𝑥10x_{1}>0 it is semiasymptotically stable from the right;

      • if x1<0subscript𝑥10x_{1}<0 it is semiasymptotically stable from the left.

  3. 3.

    If x0=x1=x2=0subscript𝑥0subscript𝑥1subscript𝑥20x_{0}=x_{1}=x_{2}=0, the zero fixed point has multiplicity of three and

    • if M>0𝑀0M>0, it is unstable;

    • if M<0𝑀0M<0, it is asymptotically stable.

Proof.

Notice that

g3′′(x)={2s(xx1)+4sx,if x2=x0=0,x104s(xx1)+2sx,if x1=x206sx,if x1=x2=0.subscriptsuperscript𝑔′′3𝑥cases2𝑠𝑥subscript𝑥14𝑠𝑥formulae-sequenceif subscript𝑥2subscript𝑥00subscript𝑥104𝑠𝑥subscript𝑥12𝑠𝑥if subscript𝑥1subscript𝑥206𝑠𝑥if subscript𝑥1subscript𝑥20g^{\prime\prime}_{3}(x)=\begin{cases}2s(x-x_{1})+4sx,&\text{if }x_{2}=x_{0}=0,\,x_{1}\neq 0\\ 4s(x-x_{1})+2sx,&\text{if }x_{1}=x_{2}\neq 0\\ 6sx,&\text{if }x_{1}=x_{2}=0.\end{cases} (4.26)

and

g3′′′(x)=6s0,subscriptsuperscript𝑔′′′3𝑥6𝑠0g^{\prime\prime\prime}_{3}(x)=6s\neq 0, (4.27)

for all cases. Therefore

  1. 1.

    If x2=x0=0,x10formulae-sequencesubscript𝑥2subscript𝑥00subscript𝑥10x_{2}=x_{0}=0,\,x_{1}\neq 0, the zero fixed point has multiplicity of two and we have that g3(0)=1subscriptsuperscript𝑔301g^{\prime}_{3}(0)=1, g3′′(0)=2sx10subscriptsuperscript𝑔′′302𝑠subscript𝑥10g^{\prime\prime}_{3}(0)=-2sx_{1}\neq 0 and g3′′′(0)=6s0subscriptsuperscript𝑔′′′306𝑠0g^{\prime\prime\prime}_{3}(0)=6s\neq 0; therefore, by theorem 1.7, the zero fixed point is an unstable fixed point. Applying theorem 1.9 we get the particular cases of semistability.

  2. 2.

    If x1=x20subscript𝑥1subscript𝑥20x_{1}=x_{2}\neq 0, this is fixed point has multiplicity of two and we have that g3(x1)=1subscriptsuperscript𝑔3subscript𝑥11g^{\prime}_{3}(x_{1})=1, g3′′(x1)=2sx10subscriptsuperscript𝑔′′3subscript𝑥12𝑠subscript𝑥10g^{\prime\prime}_{3}(x_{1})=2sx_{1}\neq 0 and g3′′′(0)=6s0subscriptsuperscript𝑔′′′306𝑠0g^{\prime\prime\prime}_{3}(0)=6s\neq 0; therefore, by theorem 1.7, this fixed point is unstable; moreover, the semistability cases are inferred from theorem 1.9 again.

  3. 3.

    If x0=x1=x2=0subscript𝑥0subscript𝑥1subscript𝑥20x_{0}=x_{1}=x_{2}=0, the zero fixed point has multiplicity of three and we have that g3(0)=1subscriptsuperscript𝑔301g^{\prime}_{3}(0)=1, g3′′(0)=0subscriptsuperscript𝑔′′300g^{\prime\prime}_{3}(0)=0 and g3′′′(0)=6s0subscriptsuperscript𝑔′′′306𝑠0g^{\prime\prime\prime}_{3}(0)=6s\neq 0; therefore, by theorem 1.7, if M>0𝑀0M>0 the zero fixed point is unstable and if M<0𝑀0M<0 it is asymptotically stable.

4.7 Cubic examples

Here we will deal with specific parametrizations for the fixed points x1subscript𝑥1x_{1} and x2subscript𝑥2x_{2} in order to clarify the above findings and to demonstrate how we can construct bifurcation diagrams with specific predetermined properties with cubic maps. We will consider M>0𝑀0M>0 unless otherwise stated explicitly.

Example 4.1.

First, consider linear parametrizations for both x1subscript𝑥1x_{1} and x2subscript𝑥2x_{2} as

x1(λ)=λ,x2(λ)=λ.formulae-sequencesubscript𝑥1𝜆𝜆subscript𝑥2𝜆𝜆x_{1}(\lambda)=-\lambda,\quad x_{2}(\lambda)=\lambda. (4.28)

The result is plotted in the lower panel of figure 4.1, where we see that the middle fixed point is x0=0subscript𝑥00x_{0}=0 always, so we expect this to be the only stable fixed point, until the separation between this and the other points breaks the stability condition and the period doubling bifurcation cascade sets on. The corresponding stability conditions are shown in the middle panel of figure 4.1, where we confirm that the curve for x0subscript𝑥0x_{0} is the only one within the stability band (2, 0)2 0(-2,\,0) until λ1.45𝜆1.45\lambda\approx 1.45 where the curve crosses the barrier of -2 and drops from then, causing x0subscript𝑥0x_{0} to bifurcate. The corresponding bifurcation diagram is shown in the upper panel of figure 4.1, where we confirm the stated above.

Refer to caption
Figure 4.1: Bifurcation diagram (upper), stability conditions plot (middle) and fixed points plot (lower) for the linear parametrization of the fixed points example of the CCM.
Example 4.2.

Next, based on the fact that the stability conditions depend upon the product of the distances between the stable fixed point and the other two fixed points, we take square root parametrizations of the fixed points and so that the product is linear. We take the parametrizations as

x1(λ)=λ,x2(λ)=λ,formulae-sequencesubscript𝑥1𝜆𝜆subscript𝑥2𝜆𝜆x_{1}(\lambda)=\sqrt{\lambda},\quad x_{2}(\lambda)=-\sqrt{\lambda}, (4.29)

The plot of the fixed points is shown in lower panel of figure 4.2, where we see again that the middle fixed point is x0subscript𝑥0x_{0}, so this will be the only attracting fixed point as long as its corresponding stability curve is within the stability region. In the middle panel of figure 4.2 we see that such stability curve goes out of said region at λ=2𝜆2\lambda=2 and x0subscript𝑥0x_{0} continues to bifurcate (period doubling) thereof. The corresponding bifurcation diagram is shown in the upper panel of figure 4.2, which is very much like the one of example 4.1 above, but looks “stretched” since the progression towards bifurcation is now linear, instead of quadratic, and given by the stability conditions curves.

Refer to caption
Figure 4.2: Bifurcation diagram (upper), stability conditions plot (middle) and fixed points plot (lower) for the square root parametrizations of the fixed points example of the CCM.
Example 4.3.

Now we will explore the full range of stability regions by making a linearly varying fixed point pass through the regions defined by the constant x0subscript𝑥0x_{0} and a constant x1subscript𝑥1x_{1}. We define then

x1(λ)=2,x2(λ)=6λ+1,formulae-sequencesubscript𝑥1𝜆2subscript𝑥2𝜆6𝜆1x_{1}(\lambda)=2,\quad x_{2}(\lambda)=6\lambda+1, (4.30)

We then obtain the plot of the lower panel of figure 4.3, for the selected range of interest of the parameter λ𝜆\lambda. In the middle panel of the same figure we can see the stability curves for the fixed points, where we see that initially, from left to right, all fixed points are unstable then, progressively, x0subscript𝑥0x_{0}, x2subscript𝑥2x_{2} and x1subscript𝑥1x_{1} become stable, the latter one losing stability for still greater values of λ𝜆\lambda. The corresponding bifurcation diagram is shown in the upper panel, where we can see how first, the stable fixed point is x0=0subscript𝑥00x_{0}=0, since it is the middle one, but begins in the chaotic region and goes “reversal” towards being stable; then, as x2subscript𝑥2x_{2} crosses through zero, it becomes the middle stable fixed point and, when it in turn crosses the constant x2subscript𝑥2x_{2}, this latter one becomes the stable fixed point, again loosing stability when x2subscript𝑥2x_{2} crosses the stability band for x1subscript𝑥1x_{1}.

Refer to caption
Figure 4.3: Bifurcation diagram (upper), stability conditions plot (middle) and fixed points plot (lower) for the constant and linear mixed parametrizations of the fixed points example of the CCM.
Example 4.4.

This example is analogous to example 3.5.5 of chapter 3, where we had the x1subscript𝑥1x_{1} fixed point of the CQM tend asymptotically to a value of approximately bsubscript𝑏-b_{\infty}. To achieve the same qualitative phenomenon in the CCM, we can define the roots as

x1(λ)=1.817eλ,x2(λ)=1.817+eλformulae-sequencesubscript𝑥1𝜆1.817superscript𝑒𝜆subscript𝑥2𝜆1.817superscript𝑒𝜆x_{1}(\lambda)=1.817-e^{-\lambda},\quad x_{2}(\lambda)=-1.817+e^{-\lambda} (4.31)

so that x2=x1subscript𝑥2subscript𝑥1x_{2}=-x_{1} and the fixed points are symmetric about the zero fixed point; their product, which gives the stability condition for the zero fixed point, will then be 3.301489+3.634eλ+e2λ3.3014893.634superscript𝑒𝜆superscript𝑒2𝜆-3.301489+3.634e^{-\lambda}+e^{-2\lambda} which will asymptotically tend to 3.301489b3.301489subscript𝑏-3.301489\approx-b_{\infty}, as λ𝜆\lambda\rightarrow\infty; this allowed us to calculate the bifurcation values of table 4.1. The lower panel of figure 4.4 shows the graphs of the fixed points, the middle panel shows the stability conditions for each fixed point and the upper panel shows the corresponding bifurcation diagram. This example allows us to determine the bifurcation values cksubscript𝑐𝑘c_{k} of the CCM with more precision; the values obtained are shown in table 4.1.

Refer to caption
Figure 4.4: Bifurcation diagram (upper), stability conditions plot (middle) and fixed points plot (lower) for the exponential asymptotic parametrizations of the fixed points example 4.4 for the CCM.

Chapter 5 n-th degree polynomial maps

Consider again a one-dimensional discrete dynamical system defined by

yn+1=f(yn;λ)subscript𝑦𝑛1𝑓subscript𝑦𝑛𝜆y_{n+1}=f(y_{n};\lambda) (5.1)

Where f𝑓f is a polynomial in one real variable y𝑦y with real fixed points and whose coefficients depend smoothly on the real parameter λ𝜆\lambda. Depending on the form of f𝑓f we have arrived previously to the definitions of the General, Linear Factors and Canonical Forms of the polynomial quadratic and cubic maps. Next, we will define precisely what we understand by the General and Canonical Maps of n𝑛n-th degree.

Definition 5.1 (General Polynomial Map).

The General Polynomial Map of n𝑛n-th degree (GPM-n𝑛n) is defined by

fn(y):=y+(1)n1Pfn(y)assignsubscript𝑓𝑛𝑦𝑦superscript1𝑛1subscript𝑃subscript𝑓𝑛𝑦f_{n}(y):=y+(-1)^{n-1}P_{f_{n}}(y) (5.2)

where

Pfn(y):=(1)n1i=0naiyi.assignsubscript𝑃subscript𝑓𝑛𝑦superscript1𝑛1superscriptsubscript𝑖0𝑛subscript𝑎𝑖superscript𝑦𝑖P_{f_{n}}(y):=(-1)^{n-1}\sum_{i=0}^{n}a_{i}\,y^{i}. (5.3)

Pfnsubscript𝑃subscript𝑓𝑛P_{f_{n}} is called the Fixed Points Polynomial associated to fnsubscript𝑓𝑛f_{n}.

It is obvious that any n𝑛n-th degree real polynomial in one variable can be put into the General Form by means of adjusting the value of the a1subscript𝑎1a_{1} coefficient properly in the fixed points polynomial. This is the broadest class of real polynomials of finite degree.

It is clear that the roots of Pfnsubscript𝑃subscript𝑓𝑛P_{f_{n}} are the fixed points of fnsubscript𝑓𝑛f_{n}. In the case of n𝑛n odd, the fundamental theorem of algebra guarantees the existence of at least one real fixed point. Let yi,i{0,1,,n1}formulae-sequencesubscript𝑦𝑖𝑖01𝑛1y_{i}\in\mathbb{C},\,i\in\{0,1,\,...,\,n-1\}, be the n𝑛n roots of Pfnsubscript𝑃subscript𝑓𝑛P_{f_{n}}, then (yyi)𝑦subscript𝑦𝑖(y-y_{i}) is a factor of Pfnsubscript𝑃subscript𝑓𝑛P_{f_{n}} by the factor theorem, therefore we can rewrite Pfnsubscript𝑃subscript𝑓𝑛P_{f_{n}} as

Pfn(y)=M(yy0)(yyn1)=Mj=0n1(yyj),M,formulae-sequencesubscript𝑃subscript𝑓𝑛𝑦𝑀𝑦subscript𝑦0𝑦subscript𝑦𝑛1𝑀superscriptsubscriptproduct𝑗0𝑛1𝑦subscript𝑦𝑗𝑀P_{f_{n}}(y)=M(y-y_{0})\cdots(y-y_{n-1})=M\prod_{j=0}^{n-1}(y-y_{j}),\quad M\in\mathbb{R}, (5.4)

and then define the

Definition 5.2 (Linear Factors Form).

Let fnsubscript𝑓𝑛f_{n} be the GPM-n𝑛n and yjsubscript𝑦𝑗y_{j}, j{0,,n1}𝑗0𝑛1j\in\{0,\,...,\,n-1\}, its n𝑛n fixed points. Then we can write

fn(y)subscript𝑓𝑛𝑦\displaystyle f_{n}(y) =y+(1)n1Mi=0n1(yyi),absent𝑦superscript1𝑛1𝑀superscriptsubscriptproduct𝑖0𝑛1𝑦subscript𝑦𝑖\displaystyle=y+(-1)^{n-1}\,M\,\prod_{i=0}^{n-1}(y-y_{i}), (5.5)
=y+(1)n1sgn(M)|M|i=0n1(yyi),absent𝑦superscript1𝑛1sgn𝑀𝑀superscriptsubscriptproduct𝑖0𝑛1𝑦subscript𝑦𝑖\displaystyle=y+(-1)^{n-1}\mathrm{sgn}(M)|M|\prod_{i=0}^{n-1}(y-y_{i}),
=y+(1)n1sM~i=0n1(yyi),absent𝑦superscript1𝑛1𝑠~𝑀superscriptsubscriptproduct𝑖0𝑛1𝑦subscript𝑦𝑖\displaystyle=y+(-1)^{n-1}s\tilde{M}\prod_{i=0}^{n-1}(y-y_{i}),

with the definitions of s𝑠s and M~~𝑀\tilde{M} used in chapters 3 and 4. We call this the Linear Factors Form of fnsubscript𝑓𝑛f_{n}.

We can directly verify that for n=2, 3𝑛23n=2,\,3 we obtain the corresponding Linear Factors Form of the Quadratic and Cubic Maps, described on pages 3.2 and 4.2, respectively. Once we know the n𝑛n fixed points of a map fnsubscript𝑓𝑛f_{n}, it is straightforward to write its linear factors form. The motivation behind the (1)n1superscript1𝑛1(-1)^{n-1} factor is that we want that, for purely aesthetic reasons, if M>0𝑀0M>0, the fixed points are real and 0<y0<y1<<yn10subscript𝑦0subscript𝑦1subscript𝑦𝑛10<y_{0}<y_{1}<\cdots<y_{n-1}, we have that fn(0)0subscriptsuperscript𝑓𝑛00f^{\prime}_{n}(0)\geq 0, which is thus accomplished.

We will now restrict this set of polynomials to those whose fixed points polynomials have only real roots, i.e. maps with real fixed points only, though not necessarily distinct. Let us make this precise by the following

Definition 5.3 (Canonical Polynomials Set).

The Canonical Polynomials Set, denoted PC[y]subscript𝑃𝐶delimited-[]𝑦P_{C}[y], is

PC[y]:={f[y]|Pf has only real roots.}assignsubscript𝑃𝐶delimited-[]𝑦conditional-set𝑓delimited-[]𝑦Pf has only real roots.P_{C}[y]:=\{f\in\mathbb{R}[y]\,|\,\text{$P_{f}$ has only real roots.}\} (5.6)

where [y]delimited-[]𝑦\mathbb{R}[y] is the set of polynomials with real coefficients on the variable y𝑦y. Likewise, PCn[y]superscriptsubscript𝑃𝐶𝑛delimited-[]𝑦P_{C}^{n}[y] denotes PC[y]n[y]subscript𝑃𝐶delimited-[]𝑦subscript𝑛delimited-[]𝑦P_{C}[y]\bigcap\mathbb{R}_{n}[y] where n[y]subscript𝑛delimited-[]𝑦\mathbb{R}_{n}[y] is the set of polynomials of degree n𝑛n with real coefficients on the variable y𝑦y.

The set PCsubscript𝑃𝐶P_{C} has been our main work ground for the analysis in this work and, as it turns out, its elements can be put in a much nicer form, easier to understand.

We can further reduce the complexity of this set of maps by means of the transformation

y=Tn(x):=sM~1n1x+y0𝑦subscript𝑇𝑛𝑥assign𝑠superscript~𝑀1𝑛1𝑥subscript𝑦0y=T_{n}(x):=s\tilde{M}^{-\frac{1}{n-1}}x+y_{0} (5.7)

where y0subscript𝑦0y_{0} is a real fixed point of the map in its linear product form. Notice that Tnsubscript𝑇𝑛T_{n} is linear, therefore it has an inverse

x=Tn1(y)=sM~1n1(yy0).𝑥superscriptsubscript𝑇𝑛1𝑦𝑠superscript~𝑀1𝑛1𝑦subscript𝑦0x=T_{n}^{-1}(y)=s\tilde{M}^{\frac{1}{n-1}}(y-y_{0}). (5.8)

Tnsubscript𝑇𝑛T_{n} is in fact a homeomorphism. We will drop the subscript n𝑛n when referring to the transformation for no specific degree. Applying this transformation to y𝑦y we reach the

Definition 5.4 (Canonical Polynomial Map).

The Canonical Polynomial Map of n𝑛n-th degree (CPM-n𝑛n) is

gn(x):=x+(1)n1snxi=1n1(xxi),n2,formulae-sequenceassignsubscript𝑔𝑛𝑥𝑥superscript1𝑛1superscript𝑠𝑛𝑥superscriptsubscriptproduct𝑖1𝑛1𝑥subscript𝑥𝑖𝑛2g_{n}(x):=x+(-1)^{n-1}\,s^{n}\,x\,\prod_{i=1}^{n-1}(x-x_{i}),\quad n\geq 2, (5.9)

where

xi=sM~1n1(yiy0),subscript𝑥𝑖𝑠superscript~𝑀1𝑛1subscript𝑦𝑖subscript𝑦0x_{i}=s\tilde{M}^{\frac{1}{n-1}}(y_{i}-y_{0}),

and yjsubscript𝑦𝑗y_{j} are the n𝑛n fixed points of the corresponding linear factors form map of n𝑛n-th degree, (at least) y0subscript𝑦0y_{0} being real.

It is clear from the definition that x0=0subscript𝑥00x_{0}=0 always. Notice also that the xisubscript𝑥𝑖x_{i} result from evaluating Tn1superscriptsubscript𝑇𝑛1T_{n}^{-1} in the corresponding yisubscript𝑦𝑖y_{i}. We can easily prove that not only does the canonical map result from applying T𝑇T, but that the canonical map is in fact T𝑇T-conjugate to the linear factors form.

Proposition 5.1.

Let fnsubscript𝑓𝑛f_{n} and Tnsubscript𝑇𝑛T_{n} and gnsubscript𝑔𝑛g_{n} as defined above, having fnsubscript𝑓𝑛f_{n} at least one real fixed point; let y0subscript𝑦0y_{0} be this real fixed point, without loss of generality. Then fnsubscript𝑓𝑛f_{n} is Tnsubscript𝑇𝑛T_{n}-conjugate to gnsubscript𝑔𝑛g_{n}.

Proof.

It is clear that Tnsubscript𝑇𝑛T_{n} is a homeomorphism since it is linear. Then, we must only prove that Tnfn=gnTnsubscript𝑇𝑛subscript𝑓𝑛subscript𝑔𝑛subscript𝑇𝑛T_{n}\circ f_{n}=g_{n}\circ T_{n}, i.e. fn(Tn(x))=Tn(gn(x))subscript𝑓𝑛subscript𝑇𝑛𝑥subscript𝑇𝑛subscript𝑔𝑛𝑥f_{n}(T_{n}(x))=T_{n}(g_{n}(x)). We then have

fn(Tn(x))subscript𝑓𝑛subscript𝑇𝑛𝑥\displaystyle f_{n}(T_{n}(x)) =fn(sM~1n1x+y0)absentsubscript𝑓𝑛𝑠superscript~𝑀1𝑛1𝑥subscript𝑦0\displaystyle=f_{n}(s\tilde{M}^{-\frac{1}{n-1}}x+y_{0}) (5.10)
=sM~1n1x+y0+(1)n1sM~i=0n1(sM~1n1x+y0yi)absent𝑠superscript~𝑀1𝑛1𝑥subscript𝑦0superscript1𝑛1𝑠~𝑀superscriptsubscriptproduct𝑖0𝑛1𝑠superscript~𝑀1𝑛1𝑥subscript𝑦0subscript𝑦𝑖\displaystyle=s\tilde{M}^{-\frac{1}{n-1}}x+y_{0}+(-1)^{n-1}\,s\tilde{M}\,\prod_{i=0}^{n-1}(s\tilde{M}^{-\frac{1}{n-1}}x+y_{0}-y_{i})
=sM~1n1x+(1)n1s2M~M~1n1xi=1n1(sM~1n1x+y0yi)+y0absent𝑠superscript~𝑀1𝑛1𝑥superscript1𝑛1superscript𝑠2~𝑀superscript~𝑀1𝑛1𝑥superscriptsubscriptproduct𝑖1𝑛1𝑠superscript~𝑀1𝑛1𝑥subscript𝑦0subscript𝑦𝑖subscript𝑦0\displaystyle=s\tilde{M}^{-\frac{1}{n-1}}x+(-1)^{n-1}\,s^{2}\tilde{M}\,\tilde{M}^{-\frac{1}{n-1}}\,x\,\prod_{i=1}^{n-1}(s\tilde{M}^{-\frac{1}{n-1}}x+y_{0}-y_{i})+y_{0}
=sM~1n1x+(1)n1sn1M~1n1xi=1n1[xsM~1n1(yiy0)]+y0absent𝑠superscript~𝑀1𝑛1𝑥superscript1𝑛1superscript𝑠𝑛1superscript~𝑀1𝑛1𝑥superscriptsubscriptproduct𝑖1𝑛1delimited-[]𝑥𝑠superscript~𝑀1𝑛1subscript𝑦𝑖subscript𝑦0subscript𝑦0\displaystyle=s\tilde{M}^{-\frac{1}{n-1}}x+(-1)^{n-1}\,s^{n-1}\tilde{M}^{-\frac{1}{n-1}}\,x\,\prod_{i=1}^{n-1}\left[x-s\tilde{M}^{\frac{1}{n-1}}(y_{i}-y_{0})\right]+y_{0}
=sM~1n1x+(1)n1sn1M1n1xi=1n1(xxi)+y0absent𝑠superscript~𝑀1𝑛1𝑥superscript1𝑛1superscript𝑠𝑛1superscript𝑀1𝑛1𝑥superscriptsubscriptproduct𝑖1𝑛1𝑥subscript𝑥𝑖subscript𝑦0\displaystyle=s\tilde{M}^{-\frac{1}{n-1}}x+(-1)^{n-1}\,s^{n-1}M^{-\frac{1}{n-1}}\,x\,\prod_{i=1}^{n-1}\left(x-x_{i}\right)+y_{0}
=sM~1n1[x+(1)n1snxi=1n1(xxi)]+y0absent𝑠superscript~𝑀1𝑛1delimited-[]𝑥superscript1𝑛1superscript𝑠𝑛𝑥superscriptsubscriptproduct𝑖1𝑛1𝑥subscript𝑥𝑖subscript𝑦0\displaystyle=s\tilde{M}^{-\frac{1}{n-1}}\left[x+(-1)^{n-1}\,s^{n}x\,\prod_{i=1}^{n-1}\left(x-x_{i}\right)\right]+y_{0}
=Tn(gn(x)).absentsubscript𝑇𝑛subscript𝑔𝑛𝑥\displaystyle=T_{n}(g_{n}(x)).

Where we have used s2=1superscript𝑠21s^{2}=1 and s=s1𝑠superscript𝑠1s=s^{-1}. ∎

This turns out to be very useful, since we know that topological conjugacy is an equivalence relation that preserves the property of chaos (see theorem 1.13 in page 1.13). This means that the analysis of stability and chaos (i.e. the “dynamics”) of real polynomial maps with real fixed points is reduced to the study of the canonical polynomial maps defined above, since we can always take any polynomial in PC[x]subscript𝑃𝐶delimited-[]𝑥P_{C}[x] to its canonical form by means of T𝑇T, determine the stability properties and then go back to the original polynomial. A commutative diagram of the conjugacy is

y=yfnfn(y)=fn(y)TT1TT1x=xgngn(x)=gn(x)commutative-diagram𝑦=𝑦superscriptsubscript𝑓𝑛subscript𝑓𝑛𝑦=subscript𝑓𝑛𝑦𝑇absentmissing-subexpressionabsentsuperscript𝑇1missing-subexpression𝑇absentmissing-subexpressionabsentsuperscript𝑇1missing-subexpressionmissing-subexpression𝑥=𝑥subscriptsubscript𝑔𝑛subscript𝑔𝑛𝑥=subscript𝑔𝑛𝑥\begin{CD}y@=y@>{f_{n}}>{}>f_{n}(y)@=f_{n}(y)\\ @A{T}A{}A@V{}V{T^{-1}}V@A{T}A{}A@V{}V{T^{-1}}V\\ x@=x@>{}>{g_{n}}>g_{n}(x)@=g_{n}(x)\end{CD} (5.11)

5.1 Stability and chaos in the canonical map of degree n

The derivative of gnsubscript𝑔𝑛g_{n}, recalling x0=0subscript𝑥00x_{0}=0 to simplify notation, is

gn(x)=1+(1)n1snj=0n1i=0,ijn1(xxi).superscriptsubscript𝑔𝑛𝑥1superscript1𝑛1superscript𝑠𝑛superscriptsubscript𝑗0𝑛1superscriptsubscriptproductformulae-sequence𝑖0𝑖𝑗𝑛1𝑥subscript𝑥𝑖g_{n}^{\prime}(x)=1+(-1)^{n-1}\,s^{n}\,\sum_{j=0}^{n-1}\prod_{i=0,\,i\neq j}^{n-1}(x-x_{i}). (5.12)

Evaluating (5.12) in the fixed point xksubscript𝑥𝑘x_{k} we get the eigenvalue function for each xksubscript𝑥𝑘x_{k}

ϕk(λ)=gn(xk(λ))subscriptitalic-ϕ𝑘𝜆superscriptsubscript𝑔𝑛subscript𝑥𝑘𝜆\displaystyle\phi_{k}(\lambda)=g_{n}^{\prime}(x_{k}(\lambda)) =1+(1)n1sni=0,ikn1(xk(λ)xi(λ))absent1superscript1𝑛1superscript𝑠𝑛superscriptsubscriptproductformulae-sequence𝑖0𝑖𝑘𝑛1subscript𝑥𝑘𝜆subscript𝑥𝑖𝜆\displaystyle=1+(-1)^{n-1}\,s^{n}\,\prod_{i=0,\,i\neq k}^{n-1}(x_{k}(\lambda)-x_{i}(\lambda)) (5.13)
=1+sni=0,ikn1(xi(λ)xk(λ)).absent1superscript𝑠𝑛superscriptsubscriptproductformulae-sequence𝑖0𝑖𝑘𝑛1subscript𝑥𝑖𝜆subscript𝑥𝑘𝜆\displaystyle=1+s^{n}\,\prod_{i=0,\,i\neq k}^{n-1}(x_{i}(\lambda)-x_{k}(\lambda)).

Then, the asymptotic stability condition |gn(xk)|<1superscriptsubscript𝑔𝑛subscript𝑥𝑘1|g_{n}^{\prime}(x_{k})|<1 implies that

2<sni=0,ikn1(xixk)<0.2superscript𝑠𝑛superscriptsubscriptproductformulae-sequence𝑖0𝑖𝑘𝑛1subscript𝑥𝑖subscript𝑥𝑘0-2<s^{n}\,\prod_{i=0,\,i\neq k}^{n-1}(x_{i}-x_{k})<0. (5.14)

From (5.14) we can recover all the stability conditions for the fixed points of the Canonical Quadratic and Cubic Maps.

which leads us to the following

Definition 5.5 (Product Distance Function).

Let gnsubscript𝑔𝑛g_{n} be the Canonical Polynomial Map of n𝑛n-th degree and x0=0,x1,,xn1subscript𝑥00subscript𝑥1subscript𝑥𝑛1x_{0}=0,\,x_{1},\,...,\,x_{n-1} its n𝑛n fixed points, all of which depend upon the parameter λ𝜆\lambda. Let xksubscript𝑥𝑘x_{k} be a real fixed point among the latter. Then

Dn,k(λ):=sni=0,ikn1(xi(λ)xk(λ)),k{0,,n1},n2,formulae-sequenceassignsubscript𝐷𝑛𝑘𝜆superscript𝑠𝑛superscriptsubscriptproductformulae-sequence𝑖0𝑖𝑘𝑛1subscript𝑥𝑖𝜆subscript𝑥𝑘𝜆formulae-sequence𝑘0𝑛1𝑛2D_{n,k}(\lambda):=s^{n}\,\prod_{i=0,\,i\neq k}^{n-1}(x_{i}(\lambda)-x_{k}(\lambda)),\quad k\in\{0,\,...,\,n-1\},\,n\geq 2, (5.15)

is called the Product Distance Function (PDF) of xksubscript𝑥𝑘x_{k}.

The definition is motivated by the fact that Dn,ksubscript𝐷𝑛𝑘D_{n,k} is a product of the distances to xksubscript𝑥𝑘x_{k} of each of the other n1𝑛1n-1 fixed points and that this quantity is fundamental in determining the stability of the fixed points. These distances are positive when xi>xksubscript𝑥𝑖subscript𝑥𝑘x_{i}>x_{k} and is negative when xi<xksubscript𝑥𝑖subscript𝑥𝑘x_{i}<x_{k}. We have stressed the dependence on the parameter λ𝜆\lambda in the definition of Dn,ksubscript𝐷𝑛𝑘D_{n,k} so that its character as a function is clear, stemming from the corresponding dependence on λ𝜆\lambda of the fixed points. In this way, the stability condition for the k𝑘k-th fixed point is reduced to

2<Dn,k(λ)<0.2subscript𝐷𝑛𝑘𝜆0-2<D_{n,k}(\lambda)<0. (5.16)

Since Dn,ksubscript𝐷𝑛𝑘D_{n,k} must be negative in order for xksubscript𝑥𝑘x_{k} to be stable as a sufficient condition, and an odd number of factors (xixk)subscript𝑥𝑖subscript𝑥𝑘(x_{i}-x_{k}) must be negative for the product in Dn,ksubscript𝐷𝑛𝑘D_{n,k} to be negative, it follows that, if n𝑛n is even or M>0𝑀0M>0, an odd number of negative factors (xixk)subscript𝑥𝑖subscript𝑥𝑘(x_{i}-x_{k}) is a sufficient condition for the hyperbolic fixed point xksubscript𝑥𝑘x_{k} to be stable; i.e. if M>0𝑀0M>0, an odd number of fixed points must lie below xksubscript𝑥𝑘x_{k} and, consequently, an even or zero (respectively, odd) number of fixed points must lie above xksubscript𝑥𝑘x_{k} if n𝑛n is even (respectively, odd). By similar arguments, we can prove

Proposition 5.2 (Necessary conditions for the stability of xksubscript𝑥𝑘x_{k}).

Let gnsubscript𝑔𝑛g_{n}, Dn,ksubscript𝐷𝑛𝑘D_{n,k} be defined as above and xksubscript𝑥𝑘x_{k} be a hyperbolic real fixed point of gnsubscript𝑔𝑛g_{n}. The following are necessary conditions for xksubscript𝑥𝑘x_{k} to be an asymptotically stable fixed point:

  • if M>0𝑀0M>0 or n𝑛n is even, an odd number of fixed points must have values lower than xksubscript𝑥𝑘x_{k}; or

  • if n𝑛n is odd and M<0𝑀0M<0, zero or an even number of fixed points must have values lower than xksubscript𝑥𝑘x_{k}.

We must remark that the above conditions are not sufficient for a fixed point to be an attractor. The sufficient condition, however, is stated as

Theorem 5.3 (Sufficient condition for the stability of xksubscript𝑥𝑘x_{k}).

Let gnsubscript𝑔𝑛g_{n}, Dn,ksubscript𝐷𝑛𝑘D_{n,k} be defined as above and xksubscript𝑥𝑘x_{k} be a hyperbolic real fixed point of gnsubscript𝑔𝑛g_{n}. Then, a necessary and sufficient condition for xksubscript𝑥𝑘x_{k} to be an attractor is that

2<Dn,k(λ)<0.2subscript𝐷𝑛𝑘𝜆0-2<D_{n,k}(\lambda)<0. (5.17)

Below the value of -2 there are other “stability bands” that lead to further period doubling bifurcations of the fixed points as they are crossed, but they must be calculated numerically and, as we have seen, depend on the degree n𝑛n of the polynomial.

5.2 Example

We will now give an example in which the above theory is applied to different degree polynomial maps.

Example 5.1 (Quartic Maps).

Using definition 5.2 for n=4𝑛4n=4, we have that

f4(y)=yM(yy0)(yy1)(yy2)(yy3).subscript𝑓4𝑦𝑦𝑀𝑦subscript𝑦0𝑦subscript𝑦1𝑦subscript𝑦2𝑦subscript𝑦3f_{4}(y)=y-M\,(y-y_{0})(y-y_{1})(y-y_{2})(y-y_{3}). (5.18)

Suppose f4subscript𝑓4f_{4} has at least one real fixed point. Without loss of generality, suppose this fixed point is y0subscript𝑦0y_{0}. Then

T4(x)=sM~1/3x+y0.subscript𝑇4𝑥𝑠superscript~𝑀13𝑥subscript𝑦0T_{4}(x)=s\tilde{M}^{-1/3}\,x+y_{0}.

Making the substitution y=T4(x)𝑦subscript𝑇4𝑥y=T_{4}(x) we can verify that we get

g4(x)=xx(xx1)(xx2)(xx3),subscript𝑔4𝑥𝑥𝑥𝑥subscript𝑥1𝑥subscript𝑥2𝑥subscript𝑥3g_{4}(x)=x-x(x-x_{1})(x-x_{2})(x-x_{3}),

where

xi=sM~1/3(yiy0),i{0, 1, 2, 3},formulae-sequencesubscript𝑥𝑖𝑠superscript~𝑀13subscript𝑦𝑖subscript𝑦0𝑖0123x_{i}=s\tilde{M}^{1/3}(y_{i}-y_{0}),\quad i\in\{0,\,1,\,2,\,3\},

The stability a real fixed point xksubscript𝑥𝑘x_{k} is given by the product distance function

D4,k(λ)=i=0,ik3(xi(λ)xk(λ)),subscript𝐷4𝑘𝜆superscriptsubscriptproductformulae-sequence𝑖0𝑖𝑘3subscript𝑥𝑖𝜆subscript𝑥𝑘𝜆D_{4,k}(\lambda)=\prod_{i=0,\,i\neq k}^{3}(x_{i}(\lambda)-x_{k}(\lambda)),

whose value must remain between minus two and zero in order for xksubscript𝑥𝑘x_{k} to be asymptotically stable; that is, if all fixed points are real,

22\displaystyle-2 <D4,0(λ)=x1x2x3absentsubscript𝐷40𝜆subscript𝑥1subscript𝑥2subscript𝑥3\displaystyle<D_{4,0}(\lambda)=x_{1}\,x_{2}\,x_{3} <0,absent0\displaystyle<0, (5.19)
22\displaystyle-2 <D4,1(λ)=x1(x2x1)(x3x1)absentsubscript𝐷41𝜆subscript𝑥1subscript𝑥2subscript𝑥1subscript𝑥3subscript𝑥1\displaystyle<D_{4,1}(\lambda)=-x_{1}(x_{2}-x_{1})(x_{3}-x_{1}) <0,absent0\displaystyle<0,
22\displaystyle-2 <D4,2(λ)=x2(x1x2)(x3x2)absentsubscript𝐷42𝜆subscript𝑥2subscript𝑥1subscript𝑥2subscript𝑥3subscript𝑥2\displaystyle<D_{4,2}(\lambda)=-x_{2}(x_{1}-x_{2})(x_{3}-x_{2}) <0,absent0\displaystyle<0,
22\displaystyle-2 <D4,3(λ)=x3(x1x3)(x2x3)absentsubscript𝐷43𝜆subscript𝑥3subscript𝑥1subscript𝑥3subscript𝑥2subscript𝑥3\displaystyle<D_{4,3}(\lambda)=-x_{3}(x_{1}-x_{3})(x_{2}-x_{3}) <0,absent0\displaystyle<0,

for 0, x1,x2subscript𝑥1subscript𝑥2x_{1},\,x_{2} and x3subscript𝑥3x_{3} to be asymptotically stable fixed points, respectively.

For example, let

x1(λ)subscript𝑥1𝜆\displaystyle x_{1}(\lambda) =λ,absent𝜆\displaystyle=\lambda, (5.20)
x2(λ)subscript𝑥2𝜆\displaystyle x_{2}(\lambda) =λ,absent𝜆\displaystyle=-\lambda,
x3(λ)subscript𝑥3𝜆\displaystyle x_{3}(\lambda) =2λ.absent2𝜆\displaystyle=2\,\lambda.

The plots of these fixed points with their corresponding parametric dependence on λ𝜆\lambda are shown in the lower panel of figure 5.1.

In light of proposition 5.2 we expect only x0subscript𝑥0x_{0} and x3subscript𝑥3x_{3} to be able to be asymptotically stable fixed points in any given range of λ𝜆\lambda. As the middle panel of figure 5.1 shows, precisely x0subscript𝑥0x_{0} and x3subscript𝑥3x_{3} are the fixed points whose product distances cross the stability band (2, 0)2 0(-2,\,0) in the range of λ𝜆\lambda being plotted. As we recall, the product distance functions are the “stability conditions” of the fixed points. As long as the product distances remain within the stability interval, the fixed points are attractors, as we can verify in the upper panel of figure 5.1; also in this last panel, we can see the two attracting fixed points at the beginning of the plotted range; then, first x3subscript𝑥3x_{3} loses its stability and gives rise to the period doubling bifurcations cascade which leads to chaotic behavior; later, zero also loses its stability and also gives rise to period doubling and chaos.

Refer to caption
Figure 5.1: Bifurcation diagram (upper), Product Distance Functions (middle) and fixed points plot of the quartic polynomial map example.

Chapter 6 An example application of the canonical map theory

We shall now consider an example application of the theory developed in chapters 3 through 5 to a couple of well known examples of discrete dynamical systems. This applications demonstrate a new method of solution for analyzing the dynamics of real polynomial maps that simplifies upon the procedures found in current literature.

6.1 The logistic map

The logistic map is the most immediate and obvious example application. We had already encountered briefly the logistic map, Lλ(x)=λx(1x)subscript𝐿𝜆𝑥𝜆𝑥1𝑥L_{\lambda}(x)=\lambda x(1-x), in subsection 1.8 of chapter 1 and again in example 2.1 of chapter 2, where we saw that the map undergoes a series of period doubling bifurcations starting at the value of λ=3𝜆3\lambda=3, ultimately achieving a chaotic nature at λ3.570𝜆3.570\lambda\approx 3.570 [Elaydi(2008), p. 47]. We will now proceed to utilize the theory developed in the previous chapters, particularly in chapter 3 and generalized in chapter 5, to this particular map.

We already know (or can easily calculate) that the fixed points of the logistic map are y0=0subscript𝑦00y_{0}=0 and y1=λ1λsubscript𝑦1𝜆1𝜆y_{1}=\frac{\lambda-1}{\lambda} [Elaydi(2008), p. 43]. With some minor algebra, we can then check that in the corresponding Linear Factors Form of the logistic map is then

hλ(x)=xλx(xλ1λ)subscript𝜆𝑥𝑥𝜆𝑥𝑥𝜆1𝜆h_{\lambda}(x)=x-\lambda x\left(x-\frac{\lambda-1}{\lambda}\right) (6.1)

where we can identify the functions of the parameter M𝑀M, y0subscript𝑦0y_{0} and y1subscript𝑦1y_{1} from definition 3.2, on page 3.2, as

s𝑠\displaystyle s =+1,absent1\displaystyle=+1, (6.2)
M~(λ)~𝑀𝜆\displaystyle\tilde{M}(\lambda) =λ,absent𝜆\displaystyle=\lambda, (6.3)
y0(λ)subscript𝑦0𝜆\displaystyle y_{0}(\lambda) =0,absent0\displaystyle=0, (6.4)
y1(λ)subscript𝑦1𝜆\displaystyle y_{1}(\lambda) =λ1λ.absent𝜆1𝜆\displaystyle=\frac{\lambda-1}{\lambda}. (6.5)

The corresponding non-zero fixed point of the canonical logistic map, x1(λ)=sM~(y1y0)subscript𝑥1𝜆𝑠~𝑀subscript𝑦1subscript𝑦0x_{1}(\lambda)=s\tilde{M}(y_{1}-y_{0}) defined in (3.12), is then simply

x1(λ)=λ1,subscript𝑥1𝜆𝜆1x_{1}(\lambda)=\lambda-1, (6.6)

from which we can state that the canonical logistic map takes the explicit form

gλ=xx(xλ+1).subscript𝑔𝜆𝑥𝑥𝑥𝜆1g_{\lambda}=x-x(x-\lambda+1). (6.7)

In order to determine the stability properties of these fixed points, both zero and nonzero, in the canonical logistic map, it is then sufficient, as we have proved in chapters 3 and 5, to observe the behavior of the Product Distance Functions (PDFs) of these fixed points, viz.

Dg,0(λ)subscript𝐷𝑔0𝜆\displaystyle D_{g,0}(\lambda) =x10=x1=λ1,absentsubscript𝑥10subscript𝑥1𝜆1\displaystyle=x_{1}-0=x_{1}=\lambda-1, (6.8)
Dg,1(λ)subscript𝐷𝑔1𝜆\displaystyle D_{g,1}(\lambda) =0x1=x1=1λ.absent0subscript𝑥1subscript𝑥11𝜆\displaystyle=0-x_{1}=-x_{1}=1-\lambda. (6.9)

By determining when these PDFs cross the stability bands whose boundaries are shown in table 3.1 we can readily determine when these fixed points are stable or unstable, when they bifurcate and when they reach any 2nsuperscript2𝑛2^{n} attracting periodic orbit for any n𝑛n, up to crossing the bsubscript𝑏b_{\infty} band. This whole process is depicted in figure 6.1. In particular, we can see from table 3.2, and again at figure 6.1, that when 1<x1<01subscript𝑥10-1<x_{1}<0, the zero fixed point is attracting since its PDF lies within the first stability band, then exchanges stability at x1=0subscript𝑥10x_{1}=0, when this last FP becomes stable and proceeds to a period doubling cascade upon its PDF, x1subscript𝑥1-x_{1}, crossing the bands defined by the bifurcation values b1subscript𝑏1-b_{1}, b2subscript𝑏2-b_{2}, etc. until reaching x1=b2.569941subscript𝑥1subscript𝑏2.569941x_{1}=b_{\infty}\approx 2.569941. This last value, by the way, agrees quite well and improves upon the approximation reported in [Elaydi(2008)] of 3.570 for the logistic map, since with the calculations of the present work λ=1+b3.569941±5×107subscript𝜆1subscript𝑏plus-or-minus3.5699415superscript107\lambda_{\infty}=1+b_{\infty}\approx 3.569941\pm 5\times 10^{-7}. Finally, since these maps are topologically conjugate and it is known that the logistic map is chaotic starting with λ=4𝜆4\lambda=4 [Elaydi(2008)], we conclude that the CQM must be so starting from x1=3subscript𝑥13x_{1}=3, which we may define as bcsubscript𝑏𝑐b_{c}, our final “bifurcation” value.

Refer to caption
Figure 6.1: Bifurcation diagram for the canonical logistic map.

6.2 Harvesting strategies

The connection of the logistic map with population models is old and well known. In the book by [Sandefur(1990)], there are a few examples of uses of second degree polynomials as recurrence functions which are used to model “harvesting” –or hunting– strategies of a population of animals. The main idea is that the population of animals grows whenever there is food and resources in the environment which, by account of its finite resources has a certain “carrying capacity”; this leads to a maximum population this environment can hold, the population growing according to the logistic model. The population may then be “harvested” –or hunted– at a certain rate yet to be specified and, depending on this rate, it is not hard to imagine that the final fate of the population of animals may be (i) extinction, if the rate is too high; (ii) steady population below the carrying capacity, if the rate is “just right”; or (iii) steady population at its maximum value dictated by the carrying capacity of the environment. A final fourth possibility –perhaps more rare– is to (iv) bring more individuals of the species from outside the system under analysis and introduce them to it, therefore making it possible for the population to surpass in number the carrying capacity of the system, but only to return naturally to the maximum value after a finite number “time-steps”. To examine this in detail consider the system defined by the recurrence relation

Δyyn+1yn=r(1yn)yn.Δ𝑦subscript𝑦𝑛1subscript𝑦𝑛𝑟1subscript𝑦𝑛subscript𝑦𝑛\Delta y\equiv y_{n+1}-y_{n}=r(1-y_{n})y_{n}. (6.10)

Here the population growth in any given period ΔyΔ𝑦\Delta y is proportional to both the initial population ynsubscript𝑦𝑛y_{n} and the difference between this and the maximum population, normalized to the value of 1. The proportion constant is the growth rate r𝑟r. After some simplification, we can rewrite system as

yn+1=(1+r)ynryn2.subscript𝑦𝑛11𝑟subscript𝑦𝑛𝑟superscriptsubscript𝑦𝑛2y_{n+1}=(1+r)y_{n}-ry_{n}^{2}. (6.11)

We have yet to add the “harvesting term(s)”, which we might do so in several ways.

If we consider a fixed rate, say each period we harvest a proportion b𝑏b of the population (in terms of the maximum) then

f(y)=(1+r)yry2b𝑓𝑦1𝑟𝑦𝑟superscript𝑦2𝑏f(y)=(1+r)y-ry^{2}-b (6.12)

is the corresponding map of this discrete dynamical system. From here we see that the Fixed Points Polynomial (FPP) of f𝑓f is111Remember fn(y)=y(1)nPf(y)subscript𝑓𝑛𝑦𝑦superscript1𝑛subscript𝑃𝑓𝑦f_{n}(y)=y-(-1)^{n}P_{f}(y) for the n𝑛n-th degree polynomial.

Pf(y)=ry2ry+b,subscript𝑃𝑓𝑦𝑟superscript𝑦2𝑟𝑦𝑏P_{f}(y)=ry^{2}-ry+b,

whose FPs we can determine to be

y±=1±14b/r2,subscript𝑦plus-or-minusplus-or-minus114𝑏𝑟2y_{\pm}=\frac{1\pm\sqrt{1-4b/r}}{2},

from which we immediately calculate the Linear Factors Form (LFF) to be

f(y)=yry(yy+)(yy).𝑓𝑦𝑦𝑟𝑦𝑦subscript𝑦𝑦subscript𝑦f(y)=y-ry(y-y_{+})(y-y_{-}). (6.13)

From this form it is also straightforward to determine that in the corresponding canonical form we have

x1(b)=r(y+y)=r(r4b).subscript𝑥1𝑏𝑟subscript𝑦subscript𝑦𝑟𝑟4𝑏x_{1}(b)=r(y_{+}-y_{-})=\sqrt{r(r-4b)}.

Both systems are related then by the linear transformations yn=y+xn/rsubscript𝑦𝑛subscript𝑦subscript𝑥𝑛𝑟y_{n}=y_{-}+x_{n}/r and xn=r(yny)subscript𝑥𝑛𝑟subscript𝑦𝑛subscript𝑦x_{n}=r(y_{n}-y_{-}). With this choice, the zero fixed point in the canonical map corresponds to ysubscript𝑦y_{-} and the nonzero fixed point to y+subscript𝑦y_{+}. To analyze the stability of this system, consider r𝑟r to be fixed and given and b𝑏b to be the parameter of this family of systems. The one immediate conclusion is that, for x1subscript𝑥1x_{1}\in\mathbb{R}, we necessarily have that x10subscript𝑥10x_{1}\geq 0. Now, real x1subscript𝑥1x_{1} implies br/4𝑏𝑟4b\leq r/4. Over the r/4𝑟4r/4 value x1subscript𝑥1x_{1} becomes complex which does not give any fixed points (but would mean “over-harvesting”). Since “harvesting” cannot be negative, it is clear b=0𝑏0b=0 corresponds to the maximum value of x1=rsubscript𝑥1𝑟x_{1}=r and that x1=0subscript𝑥10x_{1}=0 when b=r/4𝑏𝑟4b=r/4. Remember now that to analyze the stability of x1subscript𝑥1x_{1} we consider its PDF, D1=x1subscript𝐷1subscript𝑥1D_{1}=-x_{1}. Analogously, D0=x1subscript𝐷0subscript𝑥1D_{0}=x_{1}. Since the maximum value of x1subscript𝑥1x_{1} is r𝑟r and, being this the “unconstrained growth rate”, 0<r<10𝑟10<r<1, we have that r<D1<0𝑟subscript𝐷10-r<D_{1}<0, therefore putting the x1subscript𝑥1x_{1} in the stability range between b0=0subscript𝑏00b_{0}=0 and b1=2subscript𝑏12-b_{1}=-2 determined by the first stability band. This range is never left in any situation with physical meaning so we conclude that the nonzero fixed point, i.e. y+subscript𝑦y_{+} in the original system, is always the only asymptotically stable fixed point and the zero fixed point, i.e. ysubscript𝑦y_{-} in the original system, is always unstable. The only case to analyze with care is b=0𝑏0b=0 since then the two fixed points collide, but proposition 3.9 (on page 3.9) guarantees that in this case x1=0subscript𝑥10x_{1}=0 is semistable from the right.

In conclusion:

  1. 1.

    When b=r/4𝑏𝑟4b=r/4 the population faces extinction asymptotically. Over this value extinction is achieved in a finite number of steps, there not being any more fixed points.

  2. 2.

    When 0<b<r/40𝑏𝑟40<b<r/4, xnrsubscript𝑥𝑛𝑟x_{n}\rightarrow r and the population tends to y+=0.5(1+14b/r)subscript𝑦0.5114𝑏𝑟y_{+}=0.5(1+\sqrt{1-4b/r}).

  3. 3.

    When b=0𝑏0b=0, i.e. no harvesting, xnsubscript𝑥𝑛x_{n} still tends to x1=0subscript𝑥10x_{1}=0 from the right and, correspondingly, the population tends asymptotically to y+=1subscript𝑦1y_{+}=1.

6.3 Remarks on the application of the method

As somewhat instructive as the above examples may have been, the true simplifying power of the Canonical Map Theory is not yet fully met. Therefore we encourage the reader to apply this method to test the dynamics of any discrete dynamical systems family with a polynomial map depending on a single parameter, and evaluate the usefulness of the method by yourself. Families of maps with particular parametric dependencies and of cubic order will give the most useful applications for the time being, since the theory of higher degree polynomials is so far restricted to analyzing the stability of fixed points only.

Chapter 7 Conclusions and future work

Finally, lets sum up the main findings of this work and give some directions on the future work that arises from it. Basically, we can summarize the findings of this work as having successfully given conditions for the stability of the fixed points of any real polynomial map with real fixed points and that depends on a single parameter. In order to do this we have defined “canonical polynomial maps” which are topologically conjugate to any polynomial map of the same degree with real fixed points. Then, the stability of the fixed points of the canonical polynomial maps has been found to depend solely on a special function called “product distance” of a given fixed point. The values of this product distance determine the stability of the fixed point and when it bifurcates to give rise to attracting periodic orbits of period 2nsuperscript2𝑛2^{n} for all n𝑛n, and even ultimately when chaos arises through the period doubling cascade, as it passes through different “stability bands”, although the exact values and widths of these stability bands are yet to be calculated for regions of type greater than one for higher order polynomials. The latter must be done numerically.

As specific goals achieved we can remark that in this work we have

  • Generalized the existing results on parametric dependency of the coefficients of real quadratic maps for a larger set of functions.

  • Studied in detail the dynamics of general real cubic maps when the parametric dependency of its coefficients is given by continuous functions.

  • We have obtained transformations that take general quadratic and cubic maps into one whose dynamics is known and easy to study, and that are also topologically conjugate maps, so that the properties of stability and chaos are preserved.

  • Explicitly given the set of fixed points for the the proposed ’canonical’ quadratic and cubic maps.

  • Created a method that allows to understand the dynamics of cubic maps, that also includes quadratic maps.

  • Generalized the latter method to n𝑛n-th degree real polynomial maps.

  • Proposed a method to generate a large class of regular-reversal cubic maps.

  • Reproduced the know stability characteristics of the logistic map through the proposed method in a simpler way.

  • Proposed a methodology that allows to be used to create discrete dynamical systems with some prescribed bifurcation diagram.

As future work we can point out the following:

  • The numerical values of the limits of the stability bands for still higher degree canonical polynomial maps must be calculated with numerical methods.

  • Ultimately it is desired to obtain extensive tables of the bifurcation values for higher order polynomials.

  • Algebraic multiplicity of the roots of the Fixed Points Polynomial, when nonhyperbolic fixed points take place, must be analyzed for the general n-th degree Canonical Polynomial Map.

  • The case of complex fixed points must be included in detail for completeness.

  • Once the preceding is accomplished, the possibility of extending the theory to apply it to the Taylor Polynomial of any real analytical iteration function can be analyzed.

  • The power and simplicity of the proposed methodology will best be appreciated with 3rd or higher degree polynomials and when the implications on the Taylor polynomial of any non-linear map are understood.

Bibliography

  • [Banks et al.(1992)Banks, Brooks, Cairns, Davis, and Stacey] J. Banks, J. Brooks, G. Cairns, G. Davis, and P. Stacey. On Devaney’s definition of Chaos. Am. Math. Month., 99:332–334, 1992.
  • [Devaney(1989)] Robert Devaney. Introduction to Chaotic Dynamical Systems. Addison-Wesley Publishing Company, Inc., 1989.
  • [Elaydi(2000)] Saber N. Elaydi. Discrete Chaos. Chapman & Hall/CRC, 2000.
  • [Elaydi(2008)] Saber N. Elaydi. Discrete Chaos: with Applications in Science and Engineering. Chapman & Hall/CRC, Taylor & Francis Group, 2nd. edition, 2008.
  • [Feigenbaum(1978)] Michael J. Feigenbaum. Quantitative Universality for a Class of Nonlinear Transformations. Journal of Statistical Physics, 1978.
  • [Holmgren(1994)] Richard A. Holmgren. A First Course in Discrete Dynamical Systems. Springer-Verlag New York, Inc., 1994.
  • [Marsden(1974)] Jerrold E. Marsden. Elementary Classical Analysis. W. H. Freeman and Company, San Francisco, CA, USA, 1974.
  • [Protter(1998)] Murray H. Protter. Basic Elements of Real Analysis. Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA, 1998. ISBN 0-387-98479-8.
  • [Sandefur(1990)] James T. Sandefur. Discrete Dynamical Systems: Theory and Applications. Oxford University Press, 1990.
  • [Singer(1978)] David Singer. Stable orbits and bifurcation of maps of the interval. SIAM J. Appl. Math., 2(35):260–267, 1978.
  • [Solís and Jódar(2004)] Francisco J. Solís and Lucas Jódar. Quadratic Regular Reversal Maps. Discrete Dynamics in Nature and Society, 2004(2):315–323, 2004.
  • [Spiegel et al.(2009)Spiegel, Lipschutz, and Liu] Murray R. Spiegel, Seymour Lipschutz, and John Liu. Mathematical Handbook of Formulas and Tables. McGraw-Hill, third edition, 2009. doi: 10.1036/0071548556.
  • [Vellekoop and Berglund(1994)] Michel Vellekoop and Raoul Berglund. On Intervals, Transitivity = Chaos. Am. Math. Month., 101(4):353–355, 1994. URL http://links.jstor.org/sici?sici=0002-9890%28199404%29101%3A4%3C353%3AOIT%3DC%3E2.0.CO%3B2-B.
\addappheadtotoc

Appendix A Computational and numerical tools

For revision of the algebraic calculations, scripts in the wxMaxima scripting language were used, running under Ubuntu GNU/Linux 11.10. Both wxMaxima and Ubuntu are free software111“Free” as in “free speech”.. wxMaxima is a full-featured Computer Algebra System (CAS) in which some of the plots for this work were also made. Also, for most of the plots and bifurcation diagrams, a Python script shown below was used to calculate and plot most of the bifurcation diagrams presented in this work. The Python script was also ran under Ubuntu GNU/Linux and it uses numpy and matplotlib as libraries for numerical calculations and plotting, respectively.

import numpy as np
import matplotlib.pyplot as plt
import time

###################### DEFINITION OF FUNCTIONS #################################
def gn(n, xi, s=1, x0=0.1, niter=512, nplot=256): #iterates the canonical cubic map
#niter:     total number of iterations
#nplot:     number of points to return (for graphing)
#x0:        initial condition
#xi:        array containing the 3 fixed points
#n:         order of the canonical map and number of f.p.’s
#s:         sgn(M) from the original polynomial map

x = np.zeros(nplot) #here we store the orbit to be returned for graphing
xtemp = x0 #initialize orbit with initial condition
ssn=np.power(-1,n-1)*np.power(s,n) #the sign parameter

#part of the orbit that will not be graphed:
for i in range(niter-nplot):
prod=1
for j in range(n-1):
prod*=xtemp-xi[j+1]
xtemp *= (1 + ssn*prod)

x[0]=xtemp

#part of the orbit that will be graphed:
for i in range(nplot-1):
prod=1
for j in range(n-1):
prod*=x[i]-xi[j+1]
x[i+1] = x[i]*(1 + ssn*prod)

return x

def pdfn(n,xi): #Product Distance Function Values Array
#xi: vector containing the n fixed points
#k: number of fixed point whose stability we want to calculate
#n: total number of fixed points
pdf=np.zeros(n)     #array to store product distance values for each f.p.

for k in range(n):
prod=1 #initalize the product
for j in range(n): #perform product
if(j != k):
prod *= xi[j] - xi[k]
pdf[k]=prod

return pdf

def xl(i, p, lamb): #define the fixed points as a function of the parameter
#i: number of fixed point to return value
#p: NP 2-d array containing the extra parameters for the fuctional forms
#   of the fixed points. 1st index is the number of fixed point, 2nd is
#   the number of relevant parameter for that fixed point
if(i>0):
#return lamb*(3-lamb) #quadratic FP for QCM
#return p[i][1]*np.sqrt(lamb) + p[i][2] #square root fixed points
return p[i][0]*lamb + p[i][1] #linear fixed points
#return (-1)**p[i,0]*p[i,1]+(-1)**(p[i,0]+1)*np.exp(-p[i,2]*lamb)
else:
return 0 #the zeroth fixed point is always zero in value

####################### DEFINITION OF PARAMETERS ###############################
print time.clock(), "Defining parameters..."

n=4         #number of fixed points and order of the canonical map
npl=256    #number of points to graph for each value of lambda
nit=512 + npl     #total number of iterations
lmin=0     #starting value of lambda for the graph
lmax=1.2      #ending value of lambda for the graph
p=np.zeros([n,3])   #parameters for the definition of the fixed points
p[1,0]=1
p[1,1]=0
#p[1,2]=-1
p[2,0]=-1
p[2,1]=0
#p[2,2]=1
p[3,0]=2
p[3,1]=0
#p=np.array([[0,0,0],[1,0,0],[-1,0,0]]) #for the linear FPs
#p=np.array([[0,0,0],[1,0,0],[-1,0,0]]) #for the linear FPs
#p=np.array([[0,0,0],[0,2,0],[6,1,0]])   #one linear FP another const. FP
lval=np.linspace(lmin,lmax,600) #range of the parameter for the bif. diagram
midl=(lmin+lmax)/2  #middle point of the lambda interval (bif.val. estimation)
uncert=abs(lmax-lmin)/2 #uncertainty for the bif. value estimation
xi=np.zeros(n) #the zero fixed point is always zero in value
#x0 = 0.05 #initial condition
pdf=np.zeros([n,lval.size])
fps=np.zeros([n,lval.size])
lines=[’-’,’--’,’-.’]*3
linw=[2,2,2,3,3,3,4,4,4,5,5,5]

############# PRINT ESTIMATIONS TO SCREEN ######################################
print time.clock(), "Print parameter estimates..."

#we print the values found for the bifurcation
#along with its corresponding propagated uncertainties
print "l_i \in (",lmin,", ",lmax,")"
print "l_i =",midl,"+-",uncert
#print "b_i  \in (", -p[1][1]-np.exp(-lmin),",",-p[1][1]-np.exp(-lmax),")"
#print "b_i =",-p[1][1]-np.exp(-midl),"+-",midl*np.exp(-midl)*uncert

############# BEGIN PLOT #######################################################
print time.clock(), "Beginning Fixed Points plot..."

##draw the graph of the n fixed points varying with the parameter
plt.figure(1)
plt.subplot(313)
plt.grid(True)
plt.xlabel(’Parameter’)
plt.ylabel(’FPs’)
plt.xlim(lmin,lmax)
#plt.title(’Fixed Points’)
#plt.ylim(-2,2)
for k in range(n):
fps[k]=[xl(k,p,lam) for lam in lval]
plt.plot(lval,fps[k,:],’k’,label=’FP’+str(k), ls=lines[k],linewidth=linw[k])

plt.legend()

print time.clock(), "Beginning PDFs plot..."

#draw the product distance functions of each of the n FP’s
plt.subplot(312)
plt.grid(True)
#plt.xlabel(’Parameter’)
#plt.ylabel(’PDFs’)6
#plt.ylabel(’Stability Conditions’)
plt.ylabel(’PDFs’)
#plt.title(’PDFs of the Fixed Points’)
plt.xlim(lmin,lmax)
plt.ylim(-4,1)
#print bifurcation values boundaries
plt.plot(lval,np.zeros_like(lval),’k--’)
plt.plot(lval,-2*np.ones_like(lval),’k--’)
if(n==2):
plt.plot(lval,-2.449*np.ones_like(lval),’k--’)
plt.plot(lval,-2.544*np.ones_like(lval),’k--’)
plt.plot(lval,-2.5642*np.ones_like(lval),’k--’)
#plt.plot(lval,-2.56871*np.ones_like(lval),’k--’)
#plt.plot(lval,-2.56966*np.ones_like(lval),’k--’)
#plt.plot(lval,-2.569881*np.ones_like(lval),’k--’)
plt.plot(lval,-2.57*np.ones_like(lval),’k--’)

if(n==3):
plt.plot(lval,-3*np.ones_like(lval),’k--’)
plt.plot(lval,-3.236*np.ones_like(lval),’k--’)
plt.plot(lval,-3.288*np.ones_like(lval),’k--’)
plt.plot(lval,-3.29925*np.ones_like(lval),’k--’)

for j in range(lval.size):
pdf[:,j]=pdfn(n,fps[:,j])

for k in range(n):
plt.plot(lval,pdf[k,:],’k’,label=’PDF’+str(k),ls=lines[k],linewidth=linw[k])
#plt.plot(lval,pdf[k,:],’k’,label=’SC’+str(k),ls=lines[k],linewidth=linw[k])

plt.legend()

print time.clock(), "Begin bifurcation diagram plotting..."

#draw the bifurcation diagram
plt.subplot(311)
plt.grid(True)
#plt.xlabel(’Parameter’)
plt.ylabel(’Asymptotic Value’)
plt.title(’Bifurcation Diagram’)
plt.xlim(lmin,lmax)
#plt.ylim(-2, 2)
m=0

for lamb in lval: #draws the bifurcation diagram
#    for j in range(n-1): #calculate the fixed points
#        xi[j+1]= xl(j+1,p,lamb)
#
#    aux=pdfn(n,xi)  #calculate PDF of the FP’s
#    for k in range(n):
#        pdf[k][m]=aux[k] #store the values for graphing
#plt.subplot(311)
x = gn(n, fps[:,m], x0=0.9, niter=nit, nplot=npl)
plt.plot(lamb*np.ones(npl),x,’k,’)
x = gn(n, fps[:,m], x0=-0.01,niter=nit, nplot=npl)
plt.plot(lamb*np.ones(npl),x,’k,’)
m+=1

print time.clock(), "Done."

plt.show()